diff --git a/changelog b/changelog index fcde4a4..cd3dac8 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,5 @@ +20150101 tpd src/axiom-website/patches.html 20150101.03.tpd.patch +20150101 tpd src/input/wester.input absorbed and removed, yet atain 20150101 tpd src/axiom-website/patches.html 20150101.02.tpd.patch 20150101 tpd src/input/wester.input absorbed and removed 20150101 rhx src/axiom-website/patches.html 20150101.01.rhx.patch diff --git a/patch b/patch index d8706a8..38d2e12 100644 --- a/patch +++ b/patch @@ -1 +1,2 @@ -src/input/wester.input absorbed and removed +src/input/wester.input absorbed and removed, yet again + diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index d27cc51..63ff99e 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -4884,6 +4884,8 @@ buglist: bug 7278: make complains on TESTSET=notests
buglist: bug 7279: subscripting "1"::Symbol fails
20150101.02.tpd.patch src/input/wester.input absorbed and removed
+20150101.03.tpd.patch +src/input/wester.input absorbed and removed, yet again
diff --git a/src/input/wester.input.pamphlet b/src/input/wester.input.pamphlet deleted file mode 100644 index dabef09..0000000 --- a/src/input/wester.input.pamphlet +++ /dev/null @@ -1,3079 +0,0 @@ -\documentclass{article} -\usepackage{axiom} -\setlength{\textwidth}{400pt} -\begin{document} -\title{\$SPAD/src/input wester.input} -\author{Michael Wester} -\maketitle -\begin{abstract} -\end{abstract} -\eject -\tableofcontents -\eject -\begin{chunk}{*} -)set break resume -)set messages autoload off -)set streams calculate 7 -)sys rm -f wester.output -)spool wester.output -)clear all - --- ---------- Numbers ---------- ---Let's begin by playing with numbers: infinite precision integers ---S 1 of 216 -t1:=factorial(50) ---R ---R ---R (1) 30414093201713378043612608166064768844377641568960512000000000000 ---R Type: PositiveInteger ---E 1 - ---S 2 of 216 -factor(t1) ---R ---R ---R 47 22 12 8 4 3 2 2 2 ---R (2) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 ---R Type: Factored Integer ---E 2 - ---Infinite precision rational numbers ---S 3 of 216 -1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 ---R ---R ---R 4861 ---R (3) ---- ---R 2520 ---R Type: Fraction Integer ---E 3 - --- Arbitrary precision floating point numbers ---S 4 of 216 -digits(50); ---R ---R ---R Type: PositiveInteger ---E 4 - --- This number is nearly an integer ---S 5 of 216 -exp(sqrt(163.)*%pi) ---R ---R ---R (5) 26253741 2640768743.9999999999 9925007259 7198185688 9 ---R Type: Float ---E 5 - ---S 6 of 216 -digits(20); ---R ---R ---R Type: PositiveInteger ---E 6 - --- Special functions ---S 7 of 216 -besselJ(2, 1 + %i) ---R ---R ---R (7) 4.1579886943962155E-2 + 0.24739764151330637 %i ---R Type: Complex DoubleFloat ---E 7 - --- Complete decimal expansion of a rational number ---S 8 of 216 -decimal(1/7) ---R ---R ---R ______ ---R (8) 0.142857 ---R Type: DecimalExpansion ---E 8 - --- Continued fractions ---S 9 of 216 -continuedFraction(%pi) ---R ---R ---R 1 | 1 | 1 | 1 | 1 | 1 | 1 | ---R (9) 3 + +---+ + +----+ + +---+ + +-----+ + +---+ + +---+ + +---+ + ... ---R | 7 | 15 | 1 | 292 | 1 | 1 | 1 ---R Type: ContinuedFraction Integer ---E 9 - --- Simplify an expression with nested square roots ---S 10 of 216 -s1:=sqrt(2*sqrt(3) + 4) ---R ---R ---R +---------+ ---R | +-+ ---R (10) \|2\|3 + 4 ---R Type: AlgebraicNumber ---E 10 - ---S 11 of 216 -p:POLY FRAC INT:= (ratPoly(s1::Expression Integer)::SUP FRAC INT).'z ---R ---R ---R 4 2 ---R (11) z - 8z + 4 ---R Type: Polynomial Fraction Integer ---E 11 - ---S 12 of 216 -solp:=radicalSolve p ---R ---R ---R +-+ +-+ +-+ +-+ ---R (12) [z= - \|3 - 1,z= \|3 - 1,z= - \|3 + 1,z= \|3 + 1] ---R Type: List Equation Expression Integer ---E 12 - ---S 13 of 216 -rhs select (z+-> _ - real abs (complexNumeric rhs z - complexNumeric s1) < 1.E-19,solp).1 ---R ---R ---R +-+ ---R (13) \|3 + 1 ---R Type: Expression Integer ---E 13 - ---S 14 of 216 -simplify(s1) ---R ---R ---R +---------+ ---R | +-+ ---R (14) \|2\|3 + 4 ---R Type: Expression Integer ---E 14 - --- Try a more complicated example (from the Putnam exam) ---S 15 of 216 -s1:=sqrt(14 + 3*sqrt(3 + 2*sqrt(5 - 12*sqrt(3 - 2*sqrt(2))))) ---R ---R ---R +---------------------------------------+ ---R | +------------------------------+ ---R | | +----------------------+ ---R | | | +-----------+ ---R | | | | +-+ ---R (15) \|3\|2\|- 12\|- 2\|2 + 3 + 5 + 3 + 14 ---R Type: AlgebraicNumber ---E 15 - ---S 16 of 216 -p:POLY FRAC INT:= (ratPoly(s1::Expression Integer)::SUP FRAC INT).'z ---R ---R ---R (16) ---R 32 30 28 26 24 22 ---R z - 224z + 23304z - 1494304z + 66078476z - 2135811552z ---R + ---R 20 18 16 14 ---R 52170542296z - 981761299232z + 14373744925878z - 164123059536800z ---R + ---R 12 10 8 ---R 1455002985999736z - 9894174058819680z + 50472762054977900z ---R + ---R 6 4 2 ---R - 186014091485754784z + 464209556778289704z - 693994526414475104z ---R + ---R 461208414302655313 ---R Type: Polynomial Fraction Integer ---E 16 - ---S 17 of 216 -solp:=radicalSolve p ---R ---R ---R (17) ---R +--------------------------------------------+ ---R | +--------------+ +----------------+ ---R | | +----+ | +----+ ---R \|\|18\|- 23 - 54 - \|- 18\|- 23 - 54 + 28 ---R [z= -----------------------------------------------, ---R +-+ ---R \|2 ---R +--------------------------------------------+ ---R | +--------------+ +----------------+ ---R | | +----+ | +----+ ---R \|\|18\|- 23 - 54 - \|- 18\|- 23 - 54 + 28 ---R z= - -----------------------------------------------, ---R +-+ ---R \|2 ---R +--------------------------------------------+ ---R | +--------------+ +----------------+ ---R | | +----+ | +----+ ---R \|\|18\|- 23 - 54 + \|- 18\|- 23 - 54 + 28 ---R z= -----------------------------------------------, ---R +-+ ---R \|2 ---R +--------------------------------------------+ ---R | +--------------+ +----------------+ ---R | | +----+ | +----+ ---R \|\|18\|- 23 - 54 + \|- 18\|- 23 - 54 + 28 ---R z= - -----------------------------------------------, ---R +-+ ---R \|2 ---R +----------------------------------------------+ ---R | +--------------+ +----------------+ ---R | | +----+ | +----+ ---R \|- \|18\|- 23 - 54 - \|- 18\|- 23 - 54 + 28 ---R z= -------------------------------------------------, ---R +-+ ---R \|2 ---R +----------------------------------------------+ ---R | +--------------+ +----------------+ ---R | | +----+ | +----+ ---R \|- \|18\|- 23 - 54 - \|- 18\|- 23 - 54 + 28 ---R z= - -------------------------------------------------, ---R +-+ ---R \|2 ---R +----------------------------------------------+ ---R | +--------------+ +----------------+ ---R | | +----+ | +----+ ---R \|- \|18\|- 23 - 54 + \|- 18\|- 23 - 54 + 28 ---R z= -------------------------------------------------, ---R +-+ ---R \|2 ---R +----------------------------------------------+ ---R | +--------------+ +----------------+ ---R | | +----+ | +----+ +------------+ ---R \|- \|18\|- 23 - 54 + \|- 18\|- 23 - 54 + 28 | +-+ ---R z= - -------------------------------------------------, z= \|- 6\|2 + 17 , ---R +-+ ---R \|2 ---R +------------+ +----------+ +----------+ ---R | +-+ | +-+ | +-+ +-+ ---R z= - \|- 6\|2 + 17 , z= \|6\|2 + 17 , z= - \|6\|2 + 17 , z= - \|2 - 3, ---R +-+ +-+ +-+ ---R z= \|2 - 3, z= - \|2 + 3, z= \|2 + 3] ---R Type: List Equation Expression Integer ---E 17 - ---S 18 of 216 -rhs select (z+-> _ - real abs (complexNumeric rhs z - complexNumeric s1) < 1.E-19,solp).1 ---R ---R ---R +-+ ---R (18) \|2 + 3 ---R Type: Expression Integer ---E 18 - ---S 19 of 216 -simplify(s1) ---R ---R ---R +---------------------------------------+ ---R | +------------------------------+ ---R | | +----------------------+ ---R | | | +-----------+ ---R | | | | +-+ ---R (19) \|3\|2\|- 12\|- 2\|2 + 3 + 5 + 3 + 14 ---R Type: Expression Integer ---E 19 - --- Cardinal numbers ---S 20 of 216 -2*Aleph(0) - 3 ---R ---R ---R (20) Aleph(0) ---R Type: Union(CardinalNumber,...) ---E 20 - --- ---------- Statistics ---------- --- ---------- Algebra ---------- --- Numbers are nice, but symbols allow for variability---try some high school --- algebra: rational simplification ---S 21 of 216 -(x^2 - 4)/(x^2 + 4*x + 4) ---R ---R ---R x - 2 ---R (21) ----- ---R x + 2 ---R Type: Fraction Polynomial Integer ---E 21 - --- This example requires more sophistication ---S 22 of 216 -(%e^x - 1)/(%e^(x/2) + 1) ---R ---R ---R x ---R %e - 1 ---R (22) ------- ---R x ---R - ---R 2 ---R %e + 1 ---R Type: Expression Integer ---E 22 - ---S 23 of 216 -normalize(%) ---R ---R ---R x ---R - ---R 2 ---R (23) %e - 1 ---R Type: Expression Integer ---E 23 - --- Expand and factor polynomials ---S 24 of 216 -(x + 1)^20 ---R ---R ---R (24) ---R 20 19 18 17 16 15 14 13 ---R x + 20x + 190x + 1140x + 4845x + 15504x + 38760x + 77520x ---R + ---R 12 11 10 9 8 7 6 ---R 125970x + 167960x + 184756x + 167960x + 125970x + 77520x + 38760x ---R + ---R 5 4 3 2 ---R 15504x + 4845x + 1140x + 190x + 20x + 1 ---R Type: Polynomial Integer ---E 24 - ---S 25 of 216 -D(%, x) ---R ---R ---R (25) ---R 19 18 17 16 15 14 13 ---R 20x + 380x + 3420x + 19380x + 77520x + 232560x + 542640x ---R + ---R 12 11 10 9 8 7 ---R 1007760x + 1511640x + 1847560x + 1847560x + 1511640x + 1007760x ---R + ---R 6 5 4 3 2 ---R 542640x + 232560x + 77520x + 19380x + 3420x + 380x + 20 ---R Type: Polynomial Integer ---E 25 - ---S 26 of 216 -factor(%) ---R ---R ---R 19 ---R (26) 20(x + 1) ---R Type: Factored Polynomial Integer ---E 26 - ---S 27 of 216 -x^100 - 1 ---R ---R ---R 100 ---R (27) x - 1 ---R Type: Polynomial Integer ---E 27 - ---S 28 of 216 -factor(%) ---R ---R ---R (28) ---R 2 4 3 2 4 3 2 ---R (x - 1)(x + 1)(x + 1)(x - x + x - x + 1)(x + x + x + x + 1) ---R * ---R 8 6 4 2 20 15 10 5 20 15 10 5 ---R (x - x + x - x + 1)(x - x + x - x + 1)(x + x + x + x + 1) ---R * ---R 40 30 20 10 ---R (x - x + x - x + 1) ---R Type: Factored Polynomial Integer ---E 28 - --- Factor polynomials over finite fields and field extensions ---S 29 of 216 -p:= x^4 - 3*x^2 + 1 ---R ---R ---R 4 2 ---R (29) x - 3x + 1 ---R Type: Polynomial Fraction Integer ---E 29 - ---S 30 of 216 -factor(p) ---R ---R ---R 2 2 ---R (30) (x - x - 1)(x + x - 1) ---R Type: Factored Polynomial Fraction Integer ---E 30 - ---S 31 of 216 -phi:= rootOf(phi^2 - phi - 1); ---R ---R ---R Type: AlgebraicNumber ---E 31 - ---S 32 of 216 -factor(p, [phi]) ---R ---R ---R (32) (x - phi)(x - phi + 1)(x + phi - 1)(x + phi) ---R Type: Factored Polynomial AlgebraicNumber ---E 32 - ---S 33 of 216 -factor(p :: Polynomial(PrimeField(5))) ---R ---R ---R 2 2 ---R (33) (x + 2) (x + 3) ---R Type: Factored Polynomial PrimeField 5 ---E 33 - ---S 34 of 216 -expand(%) ---R ---R ---R 4 2 ---R (34) x + 2x + 1 ---R Type: Polynomial PrimeField 5 ---E 34 - --- Partial fraction decomposition ---S 35 of 216 -(x^2 + 2*x + 3)/(x^3 + 4*x^2 + 5*x + 2) ---R ---R ---R 2 ---R x + 2x + 3 ---R (35) ----------------- ---R 3 2 ---R x + 4x + 5x + 2 ---R Type: Fraction Polynomial Integer ---E 35 - ---S 36 of 216 -padicFraction(_ - partialFraction(numerator(%) :: UnivariatePolynomial(x, Fraction Integer),_ - factor(denominator(%) :: Polynomial Integer) ::_ - Factored UnivariatePolynomial(x, Fraction Integer))) ---R ---R ---R 2 2 3 ---R (36) - ----- + -------- + ----- ---R x + 1 2 x + 2 ---R (x + 1) ---R Type: PartialFraction UnivariatePolynomial(x,Fraction Integer) ---E 36 - - --- ---------- Inequalities ---------- --- ---------- Trigonometry ---------- --- Trigonometric manipulations---these are typically difficult for students ---S 37 of 216 -r:= cos(3*x)/cos(x) ---R ---R ---R cos(3x) ---R (37) ------- ---R cos(x) ---R Type: Expression Integer ---E 37 - ---S 38 of 216 -real(complexNormalize(%)) ---R ---R ---R 2 2 ---R (38) - 2sin(x) + 2cos(x) - 1 ---R Type: Expression Integer ---E 38 - ---S 39 of 216 -real(normalize(simplify(complexNormalize(r)))) ---R ---R ---R (39) 2cos(2x) - 1 ---R Type: Expression Integer ---E 39 - --- Use rewrite rules ---S 40 of 216 -sincosAngles:= rule _ - (cos((n | integer?(n)) * x) == _ - cos((n - 1)*x) * cos(x) - sin((n - 1)*x) * sin(x); _ - sin((n | integer?(n)) * x) == _ - sin((n - 1)*x) * cos(x) + cos((n - 1)*x) * sin(x) ) ---R ---R ---R (40) ---R {cos(n x) == - sin(x)sin((n - 1)x) + cos(x)cos((n - 1)x), ---R sin(n x) == cos(x)sin((n - 1)x) + cos((n - 1)x)sin(x)} ---R Type: Ruleset(Integer,Integer,Expression Integer) ---E 40 - ---S 41 of 216 -sincosAngles r ---R ---R ---R 2 2 ---R (41) - 3sin(x) + cos(x) ---R Type: Expression Integer ---E 41 - --- ---------- Determining Zero Equivalence ---------- --- The following expressions are all equal to zero ---S 42 of 216 -sqrt(997) - (997^3)^(1/6) ---R ---R ---R (42) 0 ---R Type: AlgebraicNumber ---E 42 - ---S 43 of 216 -sqrt(999983) - (999983^3)^(1/6) ---R ---R ---R (43) 0 ---R Type: AlgebraicNumber ---E 43 - ---S 44 of 216 -s1:=(2^(1/3) + 4^(1/3))^3 - 6*(2^(1/3) + 4^(1/3)) - 6 ---R ---R ---R 3+-+3+-+2 3+-+2 3+-+ 3+-+ ---R (44) 3\|2 \|4 + (3\|2 - 6)\|4 - 6\|2 ---R Type: AlgebraicNumber ---E 44 - ---S 45 of 216 -simplify(%) ---R ---R ---R 3+-+3+-+2 3+-+2 3+-+ 3+-+ ---R (45) 3\|2 \|4 + (3\|2 - 6)\|4 - 6\|2 ---R Type: Expression Integer ---E 45 - ---S 46 of 216 -p:POLY FRAC INT:= (ratPoly(s1::Expression Integer)::SUP FRAC INT).'z ---R ---R ---R 7 5 3 2 ---R (46) z - 648z + 419904z + 7558272z + 45349632z ---R Type: Polynomial Fraction Integer ---E 46 - ---S 47 of 216 -solp:=radicalSolve p ---R ---R ---R (47) [z= 0] ---R Type: List Equation Expression Integer ---E 47 - ---S 48 of 216 -rhs select (z+-> _ - real abs (complexNumeric rhs z - complexNumeric s1) < 1.E-19,solp).1 ---R ---R ---R (48) 0 ---R Type: Expression Integer ---E 48 - --- Thi49s expression is zero for x, y > 0 and n not equal to zero ---S 49 of 216 -x^(1/n)*y^(1/n) - (x*y)^(1/n) ---R ---R ---R 1 1 1 ---R - - - ---R n n n ---R (49) - (x y) + x y ---R Type: Expression Integer ---E 49 - ---S 50 of 216 -normalize(%) ---R ---R ---R (50) 0 ---R Type: Expression Integer ---E 50 - --- See Joel Moses, ``Algebraic Simplification: A Guide for the Perplexed'', --- CACM, Volume 14, Number 8, August 1971 ---S 51 of 216 -expr:= log(tan(1/2*x + %pi/4)) - asinh(tan(x)) ---R ---R ---R 2x + %pi ---R (51) log(tan(--------)) - asinh(tan(x)) ---R 4 ---R Type: Expression Integer ---E 51 - ---S 52 of 216 -complexNormalize(%) ---R ---R ---R (52) ---R - ---R log ---R +---+ 4 ---R (2x + %pi)\|- 1 ---R ---------------- ---R 4 ---R ((%e ) - 1) ---R * ---R +----------------------------------------------------+ ---R | +---+ 4 ---R | (2x + %pi)\|- 1 ---R | ---------------- ---R | 4 ---R | 4(%e ) ---R |- -------------------------------------------------- ---R | +---+ 8 +---+ 4 ---R | (2x + %pi)\|- 1 (2x + %pi)\|- 1 ---R | ---------------- ---------------- ---R | 4 4 ---R \| (%e ) - 2(%e ) + 1 ---R + ---R +---+ 4 ---R (2x + %pi)\|- 1 ---R ---------------- ---R +---+ 4 +---+ ---R - \|- 1 (%e ) - \|- 1 ---R / ---R +---+ 4 ---R (2x + %pi)\|- 1 ---R ---------------- ---R 4 ---R (%e ) - 1 ---R + ---R +---+ 2 ---R (2x + %pi)\|- 1 ---R ---------------- ---R +---+ 4 +---+ ---R - \|- 1 (%e ) + \|- 1 ---R log(--------------------------------------) ---R +---+ 2 ---R (2x + %pi)\|- 1 ---R ---------------- ---R 4 ---R (%e ) + 1 ---R Type: Expression Integer ---E 52 - --- Use a roundabout method---show that expr is a constant equal to zero ---S 53 of 216 -D(expr, x) ---R ---R ---R (53) ---R +-----------+ ---R 2x + %pi 2 | 2 2x + %pi 2 2x + %pi ---R (tan(--------) + 1)\|tan(x) + 1 - 2tan(--------)tan(x) - 2tan(--------) ---R 4 4 4 ---R --------------------------------------------------------------------------- ---R +-----------+ ---R 2x + %pi | 2 ---R 2tan(--------)\|tan(x) + 1 ---R 4 ---R Type: Expression Integer ---E 53 - ---S 54 of 216 -normalize(rootSimp(expand(simplify(%)))) ---R ---R ---R (54) 0 ---R Type: Expression Integer ---E 54 - ---S 55 of 216 -normalize(eval(expr, x = 0)) ---R ---R ---R (55) 0 ---R Type: Expression Integer ---E 55 - ---S 56 of 216 -expr:=log((2*sqrt(r) + 1)/sqrt(4*r + 4*sqrt(r) + 1)) ---R ---R ---R +-------+ ---R |cos(3x) ---R 2 |------- + 1 ---R \| cos(x) ---R (56) log(----------------------------------------) ---R +-------------------------------------+ ---R | +-------+ ---R | |cos(3x) ---R |4cos(x) |------- + 4cos(3x) + cos(x) ---R | \| cos(x) ---R |------------------------------------- ---R \| cos(x) ---R Type: Expression Integer ---E 56 - ---S 57 of 216 -D(expr, x) ---R ---R ---R (57) 0 ---R Type: Expression Integer ---E 57 - ---S 58 of 216 -eval(expr, x = 0) ---R ---R ---R (58) 0 ---R Type: Expression Integer ---E 58 - ---S 59 of 216 -(4*r + 4*sqrt(r) + 1)^(sqrt(r)/(2*sqrt(r) + 1)) _ - * (2*sqrt(r) + 1)^(1/(2*sqrt(r) + 1)) - 2*sqrt(r) - 1 ---R ---R ---R (59) ---R 1 ---R --------------- ---R +-------+ ---R |cos(3x) ---R 2 |------- + 1 ---R +-------+ \| cos(x) ---R |cos(3x) ---R (2 |------- + 1) ---R \| cos(x) ---R * ---R +-------+ ---R |cos(3x) ---R |------- ---R \| cos(x) ---R --------------- ---R +-------+ ---R |cos(3x) ---R 2 |------- + 1 ---R +-------+ \| cos(x) ---R |cos(3x) ---R 4cos(x) |------- + 4cos(3x) + cos(x) ---R \| cos(x) ---R (-------------------------------------) ---R cos(x) ---R + ---R +-------+ ---R |cos(3x) ---R - 2 |------- - 1 ---R \| cos(x) ---R Type: Expression Integer ---E 59 - ---S 60 of 216 -normalize(%) ---R ---R ---R (60) 0 ---R Type: Expression Integer ---E 60 - --- ---------- The Complex Domain ---------- --- Complex functions---separate into their real and imaginary parts ---S 61 of 216 -rectform(z) == real(z) + %i*imag(z) ---R ---R Type: Void ---E 61 - ---S 62 of 216 -rectform(log(3 + 4*%i)) ---R ---R Compiling function rectform with type Expression Complex Integer -> ---R Expression Complex Integer ---R ---R 4 ---R log(25) + 2%i atan(-) ---R 3 ---R (62) --------------------- ---R 2 ---R Type: Expression Complex Integer ---E 62 - ---S 63 of 216 -simplify(rectform(tan(x + %i*y))) ---R ---R ---R - 2y 2 - 2y ---R - 2%i cos(x)%e sin(x) + (- 2cos(x) + 1)%e + 1 ---R (63) ----------------------------------------------------- ---R - 2y 2 - 2y ---R 2cos(x)%e sin(x) + (- 2%i cos(x) + %i)%e - %i ---R Type: Expression Complex Integer ---E 63 - --- Check for branch abuse. See David R. Stoutemyer, ``Crimes and Misdemeanors --- in the Computer Algebra Trade'', Notices of the AMS, Volume 38, Number 7, --- September 1991. This first expression can simplify to sqrt(x y)/sqrt(x), --- but no further in general (consider what happens when x, y = -1). ---S 64 of 216 -sqrt(x*y*abs(z)^2) / (sqrt(x)*abs(z)) ---R ---R ---R +-----------+ ---R | 2 ---R \|x y abs(z) ---R (64) -------------- ---R +-+ ---R abs(z)\|x ---R Type: Expression Integer ---E 64 - ---S 65 of 216 -rootSimp % ---R ---R ---R +---+ ---R \|x y ---R (65) ------ ---R +-+ ---R \|x ---R Type: Expression Integer ---E 65 - --- If z = -1, sqrt(1/z) is not equal to 1/sqrt(z) ---S 66 of 216 -sqrt(1/z) - 1/sqrt(z) ---R ---R ---R +-+ ---R |1 +-+ ---R |- \|z - 1 ---R \|z ---R (66) ------------ ---R +-+ ---R \|z ---R Type: Expression Integer ---E 66 - --- If z = 3 pi i, log(exp(z)) is not equal to z ---S 67 of 216 -log(%e^z) ---R ---R ---R (67) z ---R Type: Expression Integer ---E 67 - ---S 68 of 216 -normalize(%) ---R ---R ---R (68) z ---R Type: Expression Integer ---E 68 - --- The principal value of this expression is (10 - 4 pi) i ---S 69 of 216 -log(%e^(10*%i)) ---R ---R ---R 10%i ---R (69) log(%e ) ---R Type: Expression Complex Integer ---E 69 - ---S 70 of 216 -normalize(%) ---R ---R ---R 10%i ---R (70) log(%e ) ---R Type: Expression Complex Integer ---E 70 - --- If z = pi, arctan(tan(z)) is not equal to z ---S 71 of 216 -atan(tan(z)) ---R ---R ---R (71) z ---R Type: Expression Integer ---E 71 - --- If z = 2 pi i, sqrt(exp(z)) is not equal to exp(z/2) ---S 72 of 216 -sqrt(%e^z) - %e^(z/2) ---R ---R ---R z ---R +---+ - ---R | z 2 ---R (72) \|%e - %e ---R Type: Expression Integer ---E 72 - --- ---------- Equations ---------- --- Manipulate an equation using a natural syntax ---S 73 of 216 -(x = 0)/2 + 1 ---R ---R ---R x + 2 ---R (73) -----= 1 ---R 2 ---R Type: Equation Fraction Polynomial Integer ---E 73 - --- Solve various nonlinear equations---this cubic polynomial has all real roots ---S 74 of 216 -radicalSolve(3*x^3 - 18*x^2 + 33*x - 19 = 0, x) ---R ---R ---R (74) ---R +-------------+2 +-------------+ ---R | +-+ +---+ | +-+ +---+ ---R +---+ |\|3 + \|- 1 +---+ |\|3 + \|- 1 ---R (- 3\|- 3 + 3) |------------- + (6\|- 3 + 6) |------------- - 2 ---R 3| +-+ 3| +-+ ---R \| 6\|3 \| 6\|3 ---R [x= --------------------------------------------------------------------, ---R +-------------+ ---R | +-+ +---+ ---R +---+ |\|3 + \|- 1 ---R (3\|- 3 + 3) |------------- ---R 3| +-+ ---R \| 6\|3 ---R +-------------+2 +-------------+ ---R | +-+ +---+ | +-+ +---+ ---R +---+ |\|3 + \|- 1 +---+ |\|3 + \|- 1 ---R (- 3\|- 3 - 3) |------------- + (6\|- 3 - 6) |------------- + 2 ---R 3| +-+ 3| +-+ ---R \| 6\|3 \| 6\|3 ---R x= --------------------------------------------------------------------, ---R +-------------+ ---R | +-+ +---+ ---R +---+ |\|3 + \|- 1 ---R (3\|- 3 - 3) |------------- ---R 3| +-+ ---R \| 6\|3 ---R +-------------+2 +-------------+ ---R | +-+ +---+ | +-+ +---+ ---R |\|3 + \|- 1 |\|3 + \|- 1 ---R 3 |------------- + 6 |------------- + 1 ---R 3| +-+ 3| +-+ ---R \| 6\|3 \| 6\|3 ---R x= ------------------------------------------] ---R +-------------+ ---R | +-+ +---+ ---R |\|3 + \|- 1 ---R 3 |------------- ---R 3| +-+ ---R \| 6\|3 ---R Type: List Equation Expression Integer ---E 74 - ---S 75 of 216 -map(e +-> lhs(e) = rectform(rhs(e)), %) ---R ---R Compiling function rectform with type Expression Integer -> ---R Expression Complex Integer ---R ---R (75) ---R [ ---R x = ---R +-+ %pi 2 +-+ %pi +-+ %pi ---R (\|3 - %i)sin(---) + ((- 2%i\|3 - 2)cos(---) + 4\|3 )sin(---) ---R 18 18 18 ---R + ---R +-+ %pi 2 +-+ %pi +-+ ---R (- \|3 + %i)cos(---) - 4%i\|3 cos(---) + \|3 + %i ---R 18 18 ---R / ---R +-+ %pi +-+ %pi ---R 2\|3 sin(---) - 2%i\|3 cos(---) ---R 18 18 ---R , ---R ---R x = ---R +-+ %pi 2 +-+ %pi +-+ %pi ---R (- \|3 - %i)sin(---) + ((2%i\|3 - 2)cos(---) + 4\|3 )sin(---) ---R 18 18 18 ---R + ---R +-+ %pi 2 +-+ %pi +-+ ---R (\|3 + %i)cos(---) - 4%i\|3 cos(---) - \|3 + %i ---R 18 18 ---R / ---R +-+ %pi +-+ %pi ---R 2\|3 sin(---) - 2%i\|3 cos(---) ---R 18 18 ---R , ---R ---R x = ---R %pi 2 %pi +-+ %pi %pi 2 ---R %i sin(---) + (2cos(---) + 2\|3 )sin(---) - %i cos(---) ---R 18 18 18 18 ---R + ---R +-+ %pi ---R - 2%i\|3 cos(---) - %i ---R 18 ---R / ---R +-+ %pi +-+ %pi ---R \|3 sin(---) - %i\|3 cos(---) ---R 18 18 ---R ] ---R Type: List Equation Expression Complex Integer ---E 75 - --- Some simple seeming problems can have messy answers ---S 76 of 216 -eqn:= x^4 + x^3 + x^2 + x + 1 = 0 ---R ---R ---R 4 3 2 ---R (76) x + x + x + x + 1= 0 ---R Type: Equation Polynomial Integer ---E 76 - ---S 77 of 216 -radicalSolve(eqn, x) ---R ---R ---R (77) ---R [ ---R x = ---R - ---R 2 ---R * ---R ROOT ---R +-------------------+2 ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R - 36 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R + ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R - 30 |------------------- - 40 ---R 3| +-+ ---R \| 54\|3 ---R * ---R ROOT ---R +-------------------+2 ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R 36 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R + ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R - 15 |------------------- + 40 ---R 3| +-+ ---R \| 54\|3 ---R / ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R 36 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R + ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R - 45 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R / ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R 36 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R * ---R ROOT ---R +-------------------+2 ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R 36 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R + ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R - 15 |------------------- + 40 ---R 3| +-+ ---R \| 54\|3 ---R / ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R 36 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R + ---R +---------------------------------------------------------+ ---R | +-------------------+2 +-------------------+ ---R | | +-+ +---+ | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 |- 25\|3 + 45\|- 5 ---R |36 |------------------- - 15 |------------------- + 40 ---R | 3| +-+ 3| +-+ ---R | \| 54\|3 \| 54\|3 ---R 2 |--------------------------------------------------------- - 1 ---R | +-------------------+ ---R | | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 ---R | 36 |------------------- ---R | 3| +-+ ---R \| \| 54\|3 ---R / ---R 4 ---R , ---R ---R x = ---R 2 ---R * ---R ROOT ---R +-------------------+2 +-------------------+ ---R | +-+ +---+ | +-+ +---+ ---R |- 25\|3 + 45\|- 5 |- 25\|3 + 45\|- 5 ---R - 36 |------------------- - 30 |------------------- ---R 3| +-+ 3| +-+ ---R \| 54\|3 \| 54\|3 ---R + ---R - 40 ---R * ---R +---------------------------------------------------------+ ---R | +-------------------+2 +-------------------+ ---R | | +-+ +---+ | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 |- 25\|3 + 45\|- 5 ---R |36 |------------------- - 15 |------------------- + 40 ---R | 3| +-+ 3| +-+ ---R | \| 54\|3 \| 54\|3 ---R |--------------------------------------------------------- ---R | +-------------------+ ---R | | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 ---R | 36 |------------------- ---R | 3| +-+ ---R \| \| 54\|3 ---R + ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R - 45 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R / ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R 36 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R * ---R +---------------------------------------------------------+ ---R | +-------------------+2 +-------------------+ ---R | | +-+ +---+ | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 |- 25\|3 + 45\|- 5 ---R |36 |------------------- - 15 |------------------- + 40 ---R | 3| +-+ 3| +-+ ---R | \| 54\|3 \| 54\|3 ---R |--------------------------------------------------------- ---R | +-------------------+ ---R | | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 ---R | 36 |------------------- ---R | 3| +-+ ---R \| \| 54\|3 ---R + ---R +---------------------------------------------------------+ ---R | +-------------------+2 +-------------------+ ---R | | +-+ +---+ | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 |- 25\|3 + 45\|- 5 ---R |36 |------------------- - 15 |------------------- + 40 ---R | 3| +-+ 3| +-+ ---R | \| 54\|3 \| 54\|3 ---R 2 |--------------------------------------------------------- - 1 ---R | +-------------------+ ---R | | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 ---R | 36 |------------------- ---R | 3| +-+ ---R \| \| 54\|3 ---R / ---R 4 ---R , ---R ---R x = ---R - ---R 2 ---R * ---R ROOT ---R +-------------------+2 ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R - 36 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R + ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R - 30 |------------------- - 40 ---R 3| +-+ ---R \| 54\|3 ---R * ---R ROOT ---R +-------------------+2 ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R 36 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R + ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R - 15 |------------------- + 40 ---R 3| +-+ ---R \| 54\|3 ---R / ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R 36 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R + ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R 45 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R / ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R 36 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R * ---R ROOT ---R +-------------------+2 ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R 36 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R + ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R - 15 |------------------- + 40 ---R 3| +-+ ---R \| 54\|3 ---R / ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R 36 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R + ---R +---------------------------------------------------------+ ---R | +-------------------+2 +-------------------+ ---R | | +-+ +---+ | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 |- 25\|3 + 45\|- 5 ---R |36 |------------------- - 15 |------------------- + 40 ---R | 3| +-+ 3| +-+ ---R | \| 54\|3 \| 54\|3 ---R - 2 |--------------------------------------------------------- - 1 ---R | +-------------------+ ---R | | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 ---R | 36 |------------------- ---R | 3| +-+ ---R \| \| 54\|3 ---R / ---R 4 ---R , ---R ---R x = ---R 2 ---R * ---R ROOT ---R +-------------------+2 +-------------------+ ---R | +-+ +---+ | +-+ +---+ ---R |- 25\|3 + 45\|- 5 |- 25\|3 + 45\|- 5 ---R - 36 |------------------- - 30 |------------------- ---R 3| +-+ 3| +-+ ---R \| 54\|3 \| 54\|3 ---R + ---R - 40 ---R * ---R +---------------------------------------------------------+ ---R | +-------------------+2 +-------------------+ ---R | | +-+ +---+ | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 |- 25\|3 + 45\|- 5 ---R |36 |------------------- - 15 |------------------- + 40 ---R | 3| +-+ 3| +-+ ---R | \| 54\|3 \| 54\|3 ---R |--------------------------------------------------------- ---R | +-------------------+ ---R | | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 ---R | 36 |------------------- ---R | 3| +-+ ---R \| \| 54\|3 ---R + ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R 45 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R / ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R 36 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R * ---R +---------------------------------------------------------+ ---R | +-------------------+2 +-------------------+ ---R | | +-+ +---+ | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 |- 25\|3 + 45\|- 5 ---R |36 |------------------- - 15 |------------------- + 40 ---R | 3| +-+ 3| +-+ ---R | \| 54\|3 \| 54\|3 ---R |--------------------------------------------------------- ---R | +-------------------+ ---R | | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 ---R | 36 |------------------- ---R | 3| +-+ ---R \| \| 54\|3 ---R + ---R +---------------------------------------------------------+ ---R | +-------------------+2 +-------------------+ ---R | | +-+ +---+ | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 |- 25\|3 + 45\|- 5 ---R |36 |------------------- - 15 |------------------- + 40 ---R | 3| +-+ 3| +-+ ---R | \| 54\|3 \| 54\|3 ---R - 2 |--------------------------------------------------------- - 1 ---R | +-------------------+ ---R | | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 ---R | 36 |------------------- ---R | 3| +-+ ---R \| \| 54\|3 ---R / ---R 4 ---R ] ---R Type: List Equation Expression Integer ---E 77 - --- Check one of the answers ---S 78 of 216 -eval(eqn, %.1) ---R ---R ---R (78) ---R +-------------------+ ---R | +-+ +---+ ---R +---+ +-+2 +---+ |- 25\|3 + 45\|- 5 ---R (90\|- 5 \|3 - 270\|- 5 ) |------------------- ---R 3| +-+ ---R \| 54\|3 ---R * ---R ROOT ---R +-------------------+2 +-------------------+ ---R | +-+ +---+ | +-+ +---+ ---R |- 25\|3 + 45\|- 5 |- 25\|3 + 45\|- 5 ---R (- 36 |------------------- - 30 |------------------- - 40) ---R 3| +-+ 3| +-+ ---R \| 54\|3 \| 54\|3 ---R * ---R +---------------------------------------------------------+ ---R | +-------------------+2 +-------------------+ ---R | | +-+ +---+ | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 |- 25\|3 + 45\|- 5 ---R |36 |------------------- - 15 |------------------- + 40 ---R | 3| +-+ 3| +-+ ---R | \| 54\|3 \| 54\|3 ---R |--------------------------------------------------------- ---R | +-------------------+ ---R | | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 ---R | 36 |------------------- ---R | 3| +-+ ---R \| \| 54\|3 ---R + ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R - 45 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R / ---R +-------------------+ ---R | +-+ +---+ ---R |- 25\|3 + 45\|- 5 ---R 36 |------------------- ---R 3| +-+ ---R \| 54\|3 ---R * ---R +---------------------------------------------------------+ ---R | +-------------------+2 +-------------------+ ---R | | +-+ +---+ | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 |- 25\|3 + 45\|- 5 ---R |36 |------------------- - 15 |------------------- + 40 ---R | 3| +-+ 3| +-+ ---R | \| 54\|3 \| 54\|3 ---R |--------------------------------------------------------- ---R | +-------------------+ ---R | | +-+ +---+ ---R | |- 25\|3 + 45\|- 5 ---R | 36 |------------------- ---R | 3| +-+ ---R \| \| 54\|3 ---R + ---R +-------------------+ ---R | +-+ +---+ ---R +---+ +-+2 +---+ |- 25\|3 + 45\|- 5 ---R (- 135\|- 5 \|3 + 405\|- 5 ) |------------------- ---R 3| +-+ ---R \| 54\|3 ---R / ---R +-------------------+2 ---R | +-+ +---+ ---R +---+ +-+2 +-+ |- 25\|3 + 45\|- 5 ---R (432\|- 5 \|3 + 1584\|3 ) |------------------- ---R 3| +-+ ---R \| 54\|3 ---R + ---R +-------------------+ ---R | +-+ +---+ ---R +---+ +-+2 +-+ |- 25\|3 + 45\|- 5 +-+ +---+ ---R (- 180\|- 5 \|3 - 660\|3 ) |------------------- + 1760\|3 + 1440\|- 5 ---R 3| +-+ ---R \| 54\|3 ---R = ---R 0 ---R Type: Equation Expression Integer ---E 78 - ---S 79 of 216 -%e^(2*x) + 2*%e^x + 1 = z ---R ---R ---R 2x x ---R (79) %e + 2%e + 1= z ---R Type: Equation Expression Integer ---E 79 - ---S 80 of 216 -solve(%, x) ---R ---R ---R +-+ +-+ ---R (80) [x= log(\|z - 1),x= log(- \|z - 1)] ---R Type: List Equation Expression Integer ---E 80 - --- This equation is already factored and so *should* be easy to solve ---S 81 of 216 -(x + 1) * (sin(x)^2 + 1)^2 * cos(3*x)^3 = 0 ---R ---R ---R 3 4 3 2 3 ---R (81) (x + 1)cos(3x) sin(x) + (2x + 2)cos(3x) sin(x) + (x + 1)cos(3x) = 0 ---R Type: Equation Expression Integer ---E 81 - ---S 82 of 216 -solve(%, x) ---R ---R ---R +---+ +---+ %pi ---R (82) [x= asin(\|- 1 ),x= - asin(\|- 1 ),x= ---,x= - 1] ---R 6 ---R Type: List Equation Expression Integer ---E 82 - --- The following equations have an infinite number of solutions (let n be an --- arbitrary integer): z = 0 [+ n 2 pi i] ---S 83 of 216 -solve(%e^z = 1, z) ---R ---R ---R (83) [z= 0] ---R Type: List Equation Expression Integer ---E 83 - --- x = pi/4 [+ n pi] ---S 84 of 216 -solve(sin(x) = cos(x), x) ---R ---R ---R %pi ---R (84) [x= ---] ---R 4 ---R Type: List Equation Expression Integer ---E 84 - ---S 85 of 216 -solve(tan(x) = 1, x) ---R ---R ---R %pi ---R (85) [x= ---] ---R 4 ---R Type: List Equation Expression Integer ---E 85 - --- x = 0, 0 [+ n pi, + n 2 pi] ---S 86 of 216 -solve(sin(x) = tan(x), x) ---R ---R ---R (86) [x= 0] ---R Type: List Equation Expression Integer ---E 86 - --- This equation has no solutions ---S 87 of 216 -solve(sqrt(x^2 + 1) = x - 2, x) ---R ---R ---R (87) [] ---R Type: List Equation Expression Integer ---E 87 - --- Solve a system of linear equations ---S 88 of 216 -eq1:= x + y + z = 6 ---R ---R ---R (88) z + y + x= 6 ---R Type: Equation Polynomial Integer ---E 88 - ---S 89 of 216 -eq2:= 2*x + y + 2*z = 10 ---R ---R ---R (89) 2z + y + 2x= 10 ---R Type: Equation Polynomial Integer ---E 89 - ---S 90 of 216 -eq3:= x + 3*y + z = 10 ---R ---R ---R (90) z + 3y + x= 10 ---R Type: Equation Polynomial Integer ---E 90 - --- Note that the solution is parametric ---S 91 of 216 -solve([eq1, eq2, eq3], [x, y, z]) ---R ---R ---R (91) [[x= - %CA + 4,y= 2,z= %CA]] ---R Type: List List Equation Fraction Polynomial Integer ---E 91 - --- Solve a system of nonlinear equations ---S 92 of 216 -eq1:= x^2*y + 3*y*z - 4 = 0 ---R ---R ---R 2 ---R (92) 3y z + x y - 4= 0 ---R Type: Equation Polynomial Integer ---E 92 - ---S 93 of 216 -eq2:= -3*x^2*z + 2*y^2 + 1 = 0 ---R ---R ---R 2 2 ---R (93) - 3x z + 2y + 1= 0 ---R Type: Equation Polynomial Integer ---E 93 - ---S 94 of 216 -eq3:= 2*y*z^2 - z^2 - 1 = 0 ---R ---R ---R 2 ---R (94) (2y - 1)z - 1= 0 ---R Type: Equation Polynomial Integer ---E 94 - --- Solving this by hand would be a nightmare ---S 95 of 216 -solve([eq1, eq2, eq3], [x, y, z]) ---R ---R ---R (95) ---R [[x= 1,y= 1,z= 1], [x= - 1,y= 1,z= 1], ---R 2 2 ---R [- 3z + x + 2= 0,y= - 3z + 1,3z - 2z + 1= 0], ---R ---R 4 3 2 ---R 4 3 2 2 - 18z + 24z + 21z + 12z + 3 ---R [12z - 12z - 30z + 7z + 3x = 0, y= ------------------------------, ---R 2 ---R 5 4 3 2 ---R 6z - 6z - 9z - 7z - 3z - 1= 0] ---R ] ---R Type: List List Equation Fraction Polynomial Integer ---E 95 - --- ---------- Matrix Algebra ---------- ---S 96 of 216 -m:= matrix([[a, b], [1, a*b]]) ---R ---R ---R +a b + ---R (96) | | ---R +1 a b+ ---R Type: Matrix Polynomial Integer ---E 96 - --- Invert the matrix ---S 97 of 216 -minv:= inverse(m) ---R ---R ---R + a 1 + ---R | ------ - ------ | ---R | 2 2 | ---R | a - 1 a - 1 | ---R (97) | | ---R | 1 a | ---R |- --------- ---------| ---R | 2 2 | ---R + (a - 1)b (a - 1)b+ ---R Type: Union(Matrix Fraction Polynomial Integer,...) ---E 97 - ---S 98 of 216 -m * minv ---R ---R ---R +1 0+ ---R (98) | | ---R +0 1+ ---R Type: Matrix Fraction Polynomial Integer ---E 98 - --- Define a Vandermonde matrix (useful for doing polynomial interpolations) ---S 99 of 216 -matrix([[1, 1, 1, 1 ], _ - [w, x, y, z ], _ - [w^2, x^2, y^2, z^2], _ - [w^3, x^3, y^3, z^3]]) ---R ---R ---R +1 1 1 1 + ---R | | ---R |w x y z | ---R | | ---R (99) | 2 2 2 2| ---R |w x y z | ---R | | ---R | 3 3 3 3| ---R +w x y z + ---R Type: Matrix Polynomial Integer ---E 99 - ---S 100 of 216 -determinant(%) ---R ---R ---R (100) ---R 2 2 2 2 2 3 ---R ((x - w)y + (- x + w )y + w x - w x)z ---R + ---R 3 3 3 3 3 2 ---R ((- x + w)y + (x - w )y - w x + w x)z ---R + ---R 2 2 3 3 3 2 2 3 3 2 2 2 3 ---R ((x - w )y + (- x + w )y + w x - w x )z + (- w x + w x)y ---R + ---R 3 3 2 2 3 3 2 ---R (w x - w x)y + (- w x + w x )y ---R Type: Polynomial Integer ---E 100 - --- The following formula implies a general result ---S 101 of 216 -factor(%) ---R ---R ---R (101) (x - w)(y - x)(y - w)(z - y)(z - x)(z - w) ---R Type: Factored Polynomial Integer ---E 101 - --- Compute the eigenvalues of a matrix from its characteristic polynomial ---S 102 of 216 -m:= matrix([[ 5, -3, -7], _ - [-2, 1, 2], _ - [ 2, -3, -4]]) ---R ---R ---R + 5 - 3 - 7+ ---R | | ---R (102) |- 2 1 2 | ---R | | ---R + 2 - 3 - 4+ ---R Type: Matrix Integer ---E 102 - ---S 103 of 216 -characteristicPolynomial(m, lambda) ---R ---R ---R 3 2 ---R (103) - lambda + 2lambda + 5lambda - 6 ---R Type: Polynomial Integer ---E 103 - ---S 104 of 216 -solve(% = 0, lambda) ---R ---R ---R (104) [lambda= 3,lambda= 1,lambda= - 2] ---R Type: List Equation Fraction Polynomial Integer ---E 104 - --- ---------- Tensors ---------- --- ---------- Sums and Products ---------- --- Sums: finite and infinite ---S 105 of 216 -summation(k^3, k = 1..n) ---R ---R ---R n ---R --+ 3 ---R (105) > k ---R --+ ---R k= 1 ---R Type: Expression Integer ---E 105 - ---S 106 of 216 -sum(k^3, k = 1..n) ---R ---R ---R 4 3 2 ---R n + 2n + n ---R (106) ------------- ---R 4 ---R Type: Fraction Polynomial Integer ---E 106 - ---S 107 of 216 -limit(sum(1/k^2 + 1/k^3, k = 1..n), n = %plusInfinity) ---R ---R ---R (107) "failed" ---R Type: Union("failed",...) ---E 107 - --- Products ---S 108 of 216 -product(k, k = 1..n) ---R ---R ---R n ---R ++-++ ---R (108) | | k ---R | | ---R k= 1 ---R Type: Expression Integer ---E 108 - --- ---------- Calculus ---------- --- Limits---start with a famous example ---S 109 of 216 -limit((1 + 1/n)^n, n = %plusInfinity) ---R ---R ---R (109) %e ---R Type: Union(OrderedCompletion Expression Integer,...) ---E 109 - ---S 110 of 216 -limit((1 - cos(x))/x^2, x = 0) ---R ---R ---R 1 ---R (110) - ---R 2 ---R Type: Union(OrderedCompletion Expression Integer,...) ---E 110 - --- Apply the chain rule---this is important for PDEs and many other --- applications ---S 111 of 216 -y:= operator('y); ---R ---R ---R Type: BasicOperator ---E 111 - ---S 112 of 216 -x:= operator('x); ---R ---R ---R Type: BasicOperator ---E 112 - ---S 113 of 216 -D(y(x(t)), t, 2) ---R ---R ---R , 2 ,, , ,, ---R (113) x (t) y (x(t)) + y (x(t))x (t) ---R ---R Type: Expression Integer ---E 113 - -)clear properties x y - --- ---------- Indefinite Integrals ---------- ---S 114 of 216 -1/(x^3 + 2) ---R ---R ---R 1 ---R (114) ------ ---R 3 ---R x + 2 ---R Type: Fraction Polynomial Integer ---E 114 - --- This would be very difficult to do by hand ---S 115 of 216 -integrate(%, x) ---R ---R ---R (115) ---R +-+ 2 3+-+2 3+-+ +-+ 3+-+ ---R - \|3 log(x \|4 - 2x\|4 + 4) + 2\|3 log(x\|4 + 2) ---R + ---R +-+3+-+ +-+ ---R x\|3 \|4 - \|3 ---R 6atan(----------------) ---R 3 ---R / ---R +-+3+-+ ---R 6\|3 \|4 ---R Type: Union(Expression Integer,...) ---E 115 - ---S 116 of 216 -D(%, x) ---R ---R ---R 1 ---R (116) ------ ---R 3 ---R x + 2 ---R Type: Expression Integer ---E 116 - --- This example involves several symbolic parameters ---S 117 of 216 -integrate(1/(a + b*cos(x)), x) ---R ---R ---R (117) ---R +-------+ ---R | 2 2 2 2 ---R (- a cos(x) - b)\|b - a + (- b + a )sin(x) ---R log(----------------------------------------------) ---R b cos(x) + a ---R [---------------------------------------------------, ---R +-------+ ---R | 2 2 ---R \|b - a ---R +---------+ ---R | 2 2 ---R sin(x)\|- b + a ---R 2atan(---------------------) ---R (b + a)cos(x) + b + a ---R ----------------------------] ---R +---------+ ---R | 2 2 ---R \|- b + a ---R Type: Union(List Expression Integer,...) ---E 117 - ---S 118 of 216 -map(simplify, map(f +-> D(f, x), %)) ---R ---R ---R 1 1 ---R (118) [------------,------------] ---R b cos(x) + a b cos(x) + a ---R Type: List Expression Integer ---E 118 - --- Calculus on a non-smooth (but well defined) function ---S 119 of 216 -D(abs(x), x) ---R ---R ---R abs(x) ---R (119) ------ ---R x ---R Type: Expression Integer ---E 119 - ---S 120 of 216 -integrate(abs(x), x) ---R ---R ---R x ---R ++ ---R (120) | abs(%M)d%M ---R ++ ---R Type: Union(Expression Integer,...) ---E 120 - --- Calculus on a piecewise defined function ---S 121 of 216 -a(x) == if x < 0 then -x else x ---R ---R Type: Void ---E 121 - ---S 122 of 216 -D(a(x), x) ---R ---R Compiling function a with type Variable x -> Polynomial Integer ---R ---R (122) 1 ---R Type: Polynomial Integer ---E 122 - ---S 123 of 216 -integrate(a(x), x) ---R ---R ---R 1 2 ---R (123) - x ---R 2 ---R Type: Polynomial Fraction Integer ---E 123 - -)clear properties a - - Compiled code for a has been cleared. --- The following two integrals should be equivalent. The correct solution is --- [(1 + x)^(3/2) + (1 - x)^(3/2)] / 3 ---S 124 of 216 -integrate(x/(sqrt(1 + x) + sqrt(1 - x)), x) ---R ---R ---R +-----+ +-------+ ---R (x + 1)\|x + 1 + (- x + 1)\|- x + 1 ---R (124) ------------------------------------- ---R 3 ---R Type: Union(Expression Integer,...) ---E 124 - ---S 125 of 216 -integrate((sqrt(1 + x) - sqrt(1 - x))/2, x) ---R ---R ---R +-----+ +-------+ ---R (x + 1)\|x + 1 + (- x + 1)\|- x + 1 ---R (125) ------------------------------------- ---R 3 ---R Type: Union(Expression Integer,...) ---E 125 - --- ---------- Definite Integrals ---------- --- The following two functions have a pole at zero ---S 126 of 216 -integrate(1/x, x = -1..1) ---R ---R ---R >> Error detected within library code: ---R integrate: pole in path of integration ---R ---R Continuing to read the file... ---R ---E 126 - ---S 127 of 216 -integrate(1/x^2, x = -1..1) ---R ---R ---R >> Error detected within library code: ---R integrate: pole in path of integration ---R ---R Continuing to read the file... ---R ---E 127 - --- Different branches of the square root need to be chosen in the intervals --- [0, 1] and [1, 2]. The correct results are 4/3, [4 - sqrt(8)]/3, --- [8 - sqrt(8)]/3, respectively. ---S 128 of 216 -integrate(sqrt(x + 1/x - 2), x = 0..1) ---R ---R ---R (126) potentialPole ---R Type: Union(pole: potentialPole,...) ---E 128 - ---S 129 of 216 -integrate(sqrt(x + 1/x - 2), x = 0..1, "noPole") ---R ---R ---R 4 ---R (127) - - ---R 3 ---R Type: Union(f1: OrderedCompletion Expression Integer,...) ---E 129 - ---S 130 of 216 -integrate(sqrt(x + 1/x - 2), x = 1..2) ---R ---R ---R (128) potentialPole ---R Type: Union(pole: potentialPole,...) ---E 130 - ---S 131 of 216 -integrate(sqrt(x + 1/x - 2), x = 1..2, "noPole") ---R ---R ---R +-+ ---R - 2\|2 + 4 ---R (129) ----------- ---R 3 ---R Type: Union(f1: OrderedCompletion Expression Integer,...) ---E 131 - ---S 132 of 216 -integrate(sqrt(x + 1/x - 2), x = 0..2) ---R ---R ---R (130) potentialPole ---R Type: Union(pole: potentialPole,...) ---E 132 - ---S 133 of 216 -integrate(sqrt(x + 1/x - 2), x = 0..2, "noPole") ---R ---R ---R +-+ ---R 2\|2 ---R (131) - ----- ---R 3 ---R Type: Union(f1: OrderedCompletion Expression Integer,...) ---E 133 - -)clear properties a - --- Contour integrals ---S 134 of 216 -integrate(cos(x)/(x^2 + a^2), x = %minusInfinity..%plusInfinity) ---R ---R ---R (132) potentialPole ---R Type: Union(pole: potentialPole,...) ---E 134 - ---S 135 of 216 -integrate(cos(x)/(x^2 + a^2), x = %minusInfinity..%plusInfinity, "noPole") ---R ---R ---R (133) "failed" ---R Type: Union(fail: failed,...) ---E 135 - --- Integrand with a branch point ---S 136 of 216 -integrate(t^(a - 1)/(1 + t), t = 0..%plusInfinity) ---R ---R ---R (134) potentialPole ---R Type: Union(pole: potentialPole,...) ---E 136 - ---S 137 of 216 -integrate(t^(a - 1)/(1 + t), t = 0..%plusInfinity, "noPole") ---R ---R ---R (135) "failed" ---R Type: Union(fail: failed,...) ---E 137 - --- Multiple integrals: volume of a tetrahedron ---S 138 of 216 -integrate(integrate(integrate(1, z = 0..c*(1 - x/a - y/b)), _ - y = 0..b*(1 - x/a)), _ - x = 0..a) ---R ---R ---R a b c ---R (136) ----- ---R 6 ---R Type: Union(f1: OrderedCompletion Expression Integer,...) ---E 138 - --- ---------- Series ---------- --- Taylor series---this first example comes from special relativity ---S 139 of 216 -1/sqrt(1 - (v/c)^2) ---R ---R ---R 1 ---R (137) ------------ ---R +---------+ ---R | 2 2 ---R |- v + c ---R |--------- ---R | 2 ---R \| c ---R Type: Expression Integer ---E 139 - ---S 140 of 216 -series(%, v = 0) ---R ---R ---R 1 2 3 4 5 6 8 ---R (138) 1 + --- v + --- v + ---- v + O(v ) ---R 2 4 6 ---R 2c 8c 16c ---R Type: UnivariatePuiseuxSeries(Expression Integer,v,0) ---E 140 - ---S 141 of 216 -1/%^2 ---R ---R ---R 1 2 8 ---R (139) 1 - -- v + O(v ) ---R 2 ---R c ---R Type: UnivariatePuiseuxSeries(Expression Integer,v,0) ---E 141 - ---S 142 of 216 -tsin:= series(sin(x), x = 0) ---R ---R ---R 1 3 1 5 1 7 9 ---R (140) x - - x + --- x - ---- x + O(x ) ---R 6 120 5040 ---R Type: UnivariatePuiseuxSeries(Expression Integer,x,0) ---E 142 - ---S 143 of 216 -tcos:= series(cos(x), x = 0) ---R ---R ---R 1 2 1 4 1 6 8 ---R (141) 1 - - x + -- x - --- x + O(x ) ---R 2 24 720 ---R Type: UnivariatePuiseuxSeries(Expression Integer,x,0) ---E 143 - --- Note that additional terms will be computed as needed ---S 144 of 216 -tsin/tcos ---R ---R ---R 1 3 2 5 17 7 9 ---R (142) x + - x + -- x + --- x + O(x ) ---R 3 15 315 ---R Type: UnivariatePuiseuxSeries(Expression Integer,x,0) ---E 144 - ---S 145 of 216 -series(tan(x), x = 0) ---R ---R ---R 1 3 2 5 17 7 9 ---R (143) x + - x + -- x + --- x + O(x ) ---R 3 15 315 ---R Type: UnivariatePuiseuxSeries(Expression Integer,x,0) ---E 145 - --- Look at the Taylor series around x = 1 -)set streams calculate 1 - ---S 146 of 216 -log(x)^a*exp(-b*x) ---R ---R ---R - b x a ---R (144) %e log(x) ---R Type: Expression Integer ---E 146 - ---S 147 of 216 -series(%, x = 1) ---R ---R ---R >> Error detected within library code: ---R No series expansion ---R ---R Continuing to read the file... ---R ---E 147 - -)set streams calculate 7 - --- Compare the Taylor series of two different formulations of a function ---S 148 of 216 -taylor(log(sinh(z)) + log(cosh(z + w)), z = 0) ---R ---R ---R >> Error detected within library code: ---R No Taylor expansion: logarithmic singularity ---R ---R Continuing to read the file... ---R ---E 148 - ---S 149 of 216 -% - taylor(log(sinh(z) * cosh(z + w)), z = 0) ---R ---R ---R >> Error detected within library code: ---R No Taylor expansion: logarithmic singularity ---R ---R Continuing to read the file... ---R ---E 149 - ---S 150 of 216 -series(log(sinh(z)) + log(cosh(z + w)), z = 0) ---R ---R ---R (145) ---R w 2 w 2 w 4 w 2 ---R (%e ) + 1 (%e ) - 1 (%e ) + 14(%e ) + 1 2 ---R log(----------) + log(z) + ---------- z + ---------------------- z ---R w w 2 w 4 w 2 ---R 2%e (%e ) + 1 6(%e ) + 12(%e ) + 6 ---R + ---R w 4 w 2 ---R - 4(%e ) + 4(%e ) 3 ---R ------------------------------- z ---R w 6 w 4 w 2 ---R 3(%e ) + 9(%e ) + 9(%e ) + 3 ---R + ---R w 8 w 6 w 4 w 2 ---R - (%e ) + 116(%e ) - 486(%e ) + 116(%e ) - 1 4 ---R ---------------------------------------------------- z ---R w 8 w 6 w 4 w 2 ---R 180(%e ) + 720(%e ) + 1080(%e ) + 720(%e ) + 180 ---R + ---R w 8 w 6 w 4 w 2 ---R - 4(%e ) + 44(%e ) - 44(%e ) + 4(%e ) 5 ---R ------------------------------------------------------------ z ---R w 10 w 8 w 6 w 4 w 2 ---R 15(%e ) + 75(%e ) + 150(%e ) + 150(%e ) + 75(%e ) + 15 ---R + ---R w 12 w 10 w 8 w 6 w 4 ---R (%e ) + 258(%e ) - 6537(%e ) + 16652(%e ) - 6537(%e ) ---R + ---R w 2 ---R 258(%e ) + 1 ---R / ---R w 12 w 10 w 8 w 6 w 4 ---R 2835(%e ) + 17010(%e ) + 42525(%e ) + 56700(%e ) + 42525(%e ) ---R + ---R w 2 ---R 17010(%e ) + 2835 ---R * ---R 6 ---R z ---R + ---R w 12 w 10 w 8 w 6 w 4 w 2 ---R - 8(%e ) + 456(%e ) - 2416(%e ) + 2416(%e ) - 456(%e ) + 8(%e ) ---R / ---R w 14 w 12 w 10 w 8 w 6 ---R 315(%e ) + 2205(%e ) + 6615(%e ) + 11025(%e ) + 11025(%e ) ---R + ---R w 4 w 2 ---R 6615(%e ) + 2205(%e ) + 315 ---R * ---R 7 ---R z ---R + ---R 8 ---R O(z ) ---R Type: GeneralUnivariatePowerSeries(Expression Integer,z,0) ---E 150 - ---S 151 of 216 -% - series(log(sinh(z) * cosh(z + w)), z = 0) ---R ---R ---R 15 ---R (146) O(z ) ---R Type: GeneralUnivariatePowerSeries(Expression Integer,z,0) ---E 151 - --- Power series (compute the general formula) ---S 152 of 216 -log(sin(x)/x) ---R ---R ---R sin(x) ---R (147) log(------) ---R x ---R Type: Expression Integer ---E 152 - ---S 153 of 216 -series(%, x = 0) ---R ---R ---R 1 2 1 4 1 6 1 8 10 ---R (148) - - x - --- x - ---- x - ----- x + O(x ) ---R 6 180 2835 37800 ---R Type: UnivariatePuiseuxSeries(Expression Integer,x,0) ---E 153 - ---S 154 of 216 -exp(-x)*sin(x) ---R ---R ---R - x ---R (149) %e sin(x) ---R Type: Expression Integer ---E 154 - ---S 155 of 216 -series(%, x = 0) ---R ---R ---R 2 1 3 1 5 1 6 1 7 9 ---R (150) x - x + - x - -- x + -- x - --- x + O(x ) ---R 3 30 90 630 ---R Type: UnivariatePuiseuxSeries(Expression Integer,x,0) ---E 155 - --- Derive an explicit Taylor series solution of y as a function of x from the --- following implicit relation ---S 156 of 216 -y:= operator('y); ---R ---R ---R Type: BasicOperator ---E 156 - ---S 157 of 216 -x = sin(y(x)) + cos(y(x)) ---R ---R ---R (152) x= sin(y(x)) + cos(y(x)) ---R Type: Equation Expression Integer ---E 157 - ---S 158 of 216 -seriesSolve(%, y, x = 1, 0) ---R ---R ---R >> Error detected within library code: ---R Improper initial value ---R ---R Continuing to read the file... ---R ---E 158 - -)clear properties y - --- Pade (rational function) approximation ---S 159 of 216 -pade(1, 1, taylor(exp(-x), x = 0)) ---R ---R ---R - x + 2 ---R (153) ------- ---R x + 2 ---R Type: Union(Fraction UnivariatePolynomial(x,Expression Integer),...) ---E 159 - --- ---------- Transforms ---------- --- Laplace and inverse Laplace transforms ---S 160 of 216 -laplace(cos((w - 1)*t), t, s) ---R ---R ---R s ---R (154) ---------------- ---R 2 2 ---R w - 2w + s + 1 ---R Type: Expression Integer ---E 160 - ---S 161 of 216 -inverseLaplace(%, s, t) ---R ---R ---R +-----------+ ---R | 2 ---R (155) cos(t\|w - 2w + 1 ) ---R Type: Union(Expression Integer,...) ---E 161 - --- ---------- Difference and Differential Equations ---------- --- Second order linear recurrence equation ---S 162 of 216 -r:= operator('r); ---R ---R ---R Type: BasicOperator ---E 162 - ---S 163 of 216 -r(n + 2) - 2 * r(n + 1) + r(n) = 2 ---R ---R ---R (157) r(n + 2) - 2r(n + 1) + r(n)= 2 ---R Type: Equation Expression Integer ---E 163 - ---S 164 of 216 -[%, r(0) = 1, r(1) = m] ---R ---R ---R (158) ---R [ ---R [[r(n + 2) - 2r(n + 1) + r(n),0,0], [0,r(n + 2) - 2r(n + 1) + r(n),0], ---R [0,0,r(n + 2) - 2r(n + 1) + r(n)]] ---R = ---R +2 0 0+ ---R | | ---R |0 2 0| ---R | | ---R +0 0 2+ ---R , ---R +r(0) 0 0 + +1 0 0+ +r(1) 0 0 + + 5 - 3 - 7+ ---R | | | | | | | | ---R | 0 r(0) 0 |= |0 1 0|, | 0 r(1) 0 |= |- 2 1 2 |] ---R | | | | | | | | ---R + 0 0 r(0)+ +0 0 1+ + 0 0 r(1)+ + 2 - 3 - 4+ ---R Type: List Equation SquareMatrix(3,Expression Integer) ---E 164 - -)clear properties r - --- Second order ODE with initial conditions---solve first using Laplace --- transforms ---S 165 of 216 -f:= operator('f); ---R ---R ---R Type: BasicOperator ---E 165 - ---S 166 of 216 -ode:= D(f(t), t, 2) + 4*f(t) = sin(2*t) ---R ---R ---R ,, ---R (160) f (t) + 4f(t)= sin(2t) ---R ---R Type: Equation Expression Integer ---E 166 - ---S 167 of 216 -map(e +-> laplace(e, t, s), %) ---R ---R ---R 2 , 2 ---R (161) (s + 4)laplace(f(t),t,s) - f (0) - f(0)s= ------ ---R 2 ---R s + 4 ---R Type: Equation Expression Integer ---E 167 - --- Now, solve the ODE directly ---S 168 of 216 -solve(ode, f, t = 0, [0, 0]) ---R ---R ---R sin(2t) - 2t cos(2t) ---R (162) -------------------- ---R 8 ---R Type: Union(Expression Integer,...) ---E 168 - --- First order linear ODE ---S 169 of 216 -y:= operator('y); ---R ---R ---R Type: BasicOperator ---E 169 - ---S 170 of 216 -x^2 * D(y(x), x) + 3*x*y(x) = sin(x)/x ---R ---R ---R 2 , sin(x) ---R (164) x y (x) + 3x y(x)= ------ ---R x ---R Type: Equation Expression Integer ---E 170 - ---S 171 of 216 -solve(%, y, x) ---R ---R ---R cos(x) 1 ---R (165) [particular= - ------,basis= [--]] ---R 3 3 ---R x x ---IType: Union(Record(particular: Expression Integer,... ---E 171 - --- Nonlinear ODE ---S 172 of 216 -D(y(x), x, 2) + y(x)*D(y(x), x)^3 = 0 ---R ---R ---R ,, , 3 ---R (166) y (x) + y(x)y (x) = 0 ---R ---R Type: Equation Expression Integer ---E 172 - ---S 173 of 216 -solve(%, y, x) ---R ---R ---R >> Error detected within library code: ---R getlincoeff: not an appropriate ordinary differential equation ---R ---R Continuing to read the file... ---R ---E 173 - --- A simple parametric ODE ---S 174 of 216 -D(y(x, a), x) = a*y(x, a) ---R ---R ---R (167) y (x,a)= a y(x,a) ---R ,1 ---R Type: Equation Expression Integer ---E 174 - ---S 175 of 216 -solve(%, y, x) ---R ---R ---R >> Error detected within library code: ---R parseODE: equation has order 0 ---R ---R Continuing to read the file... ---R ---E 175 - ---S 176 of 216 -D(y(x), x) = a*y(x) ---R ---R ---R , ---R (168) y (x)= a y(x) ---R ---R Type: Equation Expression Integer ---E 176 - ---S 177 of 216 -solve(%, y, x) ---R ---R ---R a x ---R (169) [particular= 0,basis= [%e ]] ---IType: Union(Record(particular: Expression Integer,... ---E 177 - --- ODE with boundary conditions. This problem has nontrivial solutions --- y(x) = A sin([pi/2 + n pi] x) for n an arbitrary integer. ---S 178 of 216 -solve(D(y(x), x, 2) + k^2*y(x) = 0, y, x) ---R ---R ---R (170) [particular= 0,basis= [cos(k x),sin(k x)]] ---IType: Union(Record(particular: Expression Integer,... ---E 178 - --- bc(%, x = 0, y = 0, x = 1, D(y(x), x) = 0) --- System of two linear, constant coefficient ODEs ---S 179 of 216 -x:= operator('x); ---R ---R ---R Type: BasicOperator ---E 179 - ---S 180 of 216 -system:= [D(x(t), t) = x(t) - y(t), D(y(t), t) = x(t) + y(t)] ---R ---R ---R , , ---R (172) [x (t)= - y(t) + x(t),y (t)= y(t) + x(t)] ---R ---R Type: List Equation Expression Integer ---E 180 - ---S 181 of 216 -solve(system,[x,y],t) ---R ---R ---R (173) ---R t t t t ---R [particular= [0,0],basis= [[cos(t)%e ,%e sin(t)],[%e sin(t),- cos(t)%e ]]] ---IType: Union(Record(particular: Vector Expression Integer,... ---E 181 - --- Check the answer --- Triangular system of two ODEs ---S 182 of 216 -system:= [D(x(t), t) = x(t) * (1 + cos(t)/(2 + sin(t))), _ - D(y(t), t) = x(t) - y(t)] ---R ---R ---R , x(t)sin(t) + x(t)cos(t) + 2x(t) , ---R (174) [x (t)= -------------------------------,y (t)= - y(t) + x(t)] ---R sin(t) + 2 ---R Type: List Equation Expression Integer ---E 182 - --- Try solving this system one equation at a time ---S 183 of 216 -solve(system.1, x, t) ---R ---R ---R t t ---R (175) [particular= 0,basis= [%e sin(t) + 2%e ]] ---IType: Union(Record(particular: Expression Integer,... ---E 183 - ---S 184 of 216 -genericx:=C1*%.basis.1 ---R ---R ---R t t ---R (176) C1 %e sin(t) + 2C1 %e ---R Type: Expression Integer ---E 184 - ---S 185 of 216 -eval(lhs rightZero system.2,x,genericx,t) ---R ---R Compiling function %DP with type Expression Integer -> Expression ---R Integer ---R ---R , t t ---R (177) y (t) - C1 %e sin(t) - 2C1 %e + y(t) ---R ---R Type: Expression Integer ---E 185 - ---S 186 of 216 -solve(%,y,t) ---R ---R ---R (178) ---R - t t 2 - t t 2 ---R 2C1 %e (%e ) sin(t) + (- C1 cos(t) + 5C1)%e (%e ) ---R [particular= ------------------------------------------------------, ---R 5 ---R - t ---R basis= [%e ]] ---IType: Union(Record(particular: Expression Integer,... ---E 186 - ---S 187 of 216 -genericy:=simplify (%.particular)+K1*(%.basis.1) ---R ---R ---R t t - t ---R 2C1 %e sin(t) + (- C1 cos(t) + 5C1)%e + 5K1 %e ---R (179) -------------------------------------------------- ---R 5 ---R Type: Expression Integer ---E 187 - ---S 188 of 216 -eval(lhs rightZero system.1,x,genericx,t) ---R ---R Compiling function %DS with type Expression Integer -> Expression ---R Integer ---R ---R (180) 0 ---R Type: Expression Integer ---E 188 - ---S 189 of 216 -eval(lhs rightZero system.2,[x,y],[genericx,genericy],t) ---R ---R Compiling function %DT with type Expression Integer -> Expression ---R Integer ---R Compiling function %DU with type Expression Integer -> Expression ---R Integer ---R ---R (181) 0 ---R Type: Expression Integer ---E 189 - -)clear properties x y - --- ---------- Operators ---------- --- Linear differential operator ---S 190 of 216 -DD:= operator("D") :: Operator(Expression Integer) ---R ---R ---R (182) D ---R Type: Operator Expression Integer ---E 190 - ---S 191 of 216 -evaluate(DD, e +-> D(e, x))$Operator(Expression Integer) ---R ---R ---R (183) D ---R Type: Operator Expression Integer ---E 191 - ---S 192 of 216 -L:= (DD - 1) * (DD + 2) ---R ---R ---R 2 ---R (184) D 2 + D - D - 2 ---R Type: Operator Expression Integer ---E 192 - ---S 193 of 216 -g:= operator('g) ---R ---R ---R (185) g ---R Type: BasicOperator ---E 193 - ---S 194 of 216 -L(f(x)) ---R ---R ---R ,, , ---R (186) f (x) + f (x) - 2f(x) ---R ---R Type: Expression Integer ---E 194 - ---S 195 of 216 -subst(L(subst(g(y), y = x)), x = y) ---R ---R ---R ,, , ---R (187) g (y) + g (y) - 2g(y) ---R ---R Type: Expression Integer ---E 195 - ---S 196 of 216 -subst(L(subst(A * sin(z^2), z = x)), x = z) ---R ---R ---R 2 2 2 ---R (188) (- 4A z - 2A)sin(z ) + (2A z + 2A)cos(z ) ---R Type: Expression Integer ---E 196 - --- Truncated Taylor series operator ---S 197 of 216 -T:= (f, xx, a) +-> subst((DD^0)(f(x)), x = a)/factorial(0) * (xx - a)^0 + _ - subst((DD^1)(f(x)), x = a)/factorial(1) * (xx - a)^1 + _ - subst((DD^2)(f(x)), x = a)/factorial(2) * (xx - a)^2 ---R ---R ---R (189) ---R (f,xx,a) ---R +-> ---R 0 1 ---R subst(DD (f(x)),x= a) 0 subst(DD (f(x)),x= a) 1 ---R --------------------- (xx - a) + --------------------- (xx - a) ---R factorial(0) factorial(1) ---R + ---R 2 ---R subst(DD (f(x)),x= a) 2 ---R --------------------- (xx - a) ---R factorial(2) ---R Type: AnonymousFunction ---E 197 - ---S 198 of 216 -T(f, x, a) ---R ---R ---R 2 2 ,, , ---R (x - 2a x + a )f (a) + (2x - 2a)f (a) + 2f(a) ---R ---R (190) ----------------------------------------------- ---R 2 ---R Type: Expression Integer ---E 198 - ---S 199 of 216 -T(g, y, b) ---R ---R ---R 2 2 ,, , ---R (y - 2b y + b )g (b) + (2y - 2b)g (b) + 2g(b) ---R ---R (191) ----------------------------------------------- ---R 2 ---R Type: Expression Integer ---E 199 - ---S 200 of 216 -Sin:= operator("sin") :: Operator(Expression Integer) ---R ---R ---R (192) sin ---R Type: Operator Expression Integer ---E 200 - ---S 201 of 216 -evaluate(Sin, x +-> sin(x))$Operator(Expression Integer) ---R ---R ---R (193) sin ---R Type: Operator Expression Integer ---E 201 - ---S 202 of 216 -T(Sin, z, c) ---R ---R ---R 2 2 ---R (- z + 2c z - c + 2)sin(c) + (2z - 2c)cos(c) ---R (194) ---------------------------------------------- ---R 2 ---R Type: Expression Integer ---E 202 - --- ---------- Programming ---------- --- Write a simple program to compute Legendre polynomials -)clear properties p - ---S 203 of 216 -p(n, x) == 1/(2^n*factorial(n)) * D((x^2 - 1)^n, x, n) ---R ---R Type: Void ---E 203 - ---S 204 of 216 -for i in 0..4 repeat { output(""); _ - output(concat(["p(", string(i), ", x) = "])); _ - output(p(i, x))} ---R ---R Compiling function p with type (NonNegativeInteger,Variable x) -> ---R Polynomial Fraction Integer ---R ---R p(0, x) = ---R 1 ---R ---R p(1, x) = ---R x ---R ---R p(2, x) = ---R 3 2 1 ---R - x - - ---R 2 2 ---R ---R p(3, x) = ---R 5 3 3 ---R - x - - x ---R 2 2 ---R ---R p(4, x) = ---R 35 4 15 2 3 ---R -- x - -- x + - ---R 8 4 8 ---R Type: Void ---E 204 - ---S 205 of 216 -eval(p(4, x), x = 1) ---R ---R Compiling function p with type (PositiveInteger,Variable x) -> ---R Polynomial Fraction Integer ---R ---R (197) 1 ---R Type: Polynomial Fraction Integer ---E 205 - --- Now, perform the same computation using a recursive definition ---S 206 of 216 -pp(0, x) == 1 ---R ---R Type: Void ---E 206 - ---S 207 of 216 -pp(1, x) == x ---R ---R Type: Void ---E 207 - ---S 208 of 216 -pp(n, x) == ((2*n - 1)*x*pp(n - 1, x) - (n - 1)*pp(n - 2, x))/n ---R ---R Type: Void ---E 208 - ---S 209 of 216 -for i in 0..4 repeat { output(""); _ - output(concat(["pp(", string(i), ", x) = "])); _ - output(pp(i, x))} ---R ---R Compiling function pp with type (Integer,Variable x) -> Polynomial ---R Fraction Integer ---R ---R pp(0, x) = ---R 1 ---R ---R pp(1, x) = ---R x ---R ---R pp(2, x) = ---R 3 2 1 ---R - x - - ---R 2 2 ---R ---R pp(3, x) = ---R 5 3 3 ---R - x - - x ---R 2 2 ---R ---R pp(4, x) = ---R 35 4 15 2 3 ---R -- x - -- x + - ---R 8 4 8 ---R Type: Void ---E 209 - -)clear properties p pp - - Compiled code for p has been cleared. - Compiled code for pp has been cleared. --- ---------- Translation ---------- --- Horner's rule---this is important for numerical algorithms ---S 210 of 216 -a:= operator('a) ---R ---R ---R (202) a ---R Type: BasicOperator ---E 210 - ---S 211 of 216 -sum(a(i)*x^i, i = 1..5) ---R ---R ---R 5 4 3 2 ---R (203) a(5)x + a(4)x + a(3)x + a(2)x + a(1)x ---R Type: Expression Integer ---E 211 - -)clear properties p - ---S 212 of 216 -p:= factor(%) ---R ---R ---R 5 4 3 2 ---R (204) a(5)x + a(4)x + a(3)x + a(2)x + a(1)x ---R Type: Factored Expression Integer ---E 212 - --- Convert the result into FORTRAN syntax -)set fortran ints2floats off - ---S 213 of 216 -outputAsFortran('p = p) ---R ---R p=a(5)*x**5+a(4)*x**4+a(3)*x**3+a(2)*x*x+a(1)*x ---R Type: Void ---E 213 - --- ---------- Boolean Logic ---------- --- Simplify logical expressions ---S 214 of 216 -true and false ---R ---R ---R (206) false ---R Type: Boolean ---E 214 - ---S 215 of 216 -x or (not x) ---R ---R ---R Argument number 1 to "or" must be a Boolean. ---E 215 - ---S 216 of 216 -x or y or (x and y) ---R ---R ---R Argument number 1 to "or" must be a Boolean. ---E 216 - - -)spool - - -)lisp (bye) -\end{chunk} -\eject -\begin{thebibliography}{99} -\bibitem{1} nothing -\end{thebibliography} -\end{document}