#1: " ----------[ D e r i v e ]---------- " User #2: " ---------- Initialization ---------- " User #3: " ---------- Definite Integrals ---------- " User User #4: " The following two functions have a pole at a. The first integral has a " #5: " principal value of zero; the second is divergent " User a + 1 / 1 #6: | ------- dx User / x - a a - 1 #7: - pi*#i Simp(#6) a + 1 / 1 | ---------- dx #8: | 2 User / (x - a) a - 1 #9: -2 Simp(#8) User #10: " Different branches of the square root need to be chosen in the intervals " User #11: " [0, 1] and [1, 2]. The correct results are 4/3, [4 - sqrt(8)]/3, " #12: " [8 - sqrt(8)]/3, respectively " User 1 / / 1 \ #13: | SQRT|x + --- - 2| dx User / \ x / 0 4 #14: --- Simp(#13) 3 2 / / 1 \ #15: | SQRT|x + --- - 2| dx User / \ x / 1 4 2*SQRT(2) #16: --- - ----------- Simp(#15) 3 3 2 / / 1 \ #17: | SQRT|x + --- - 2| dx User / \ x / 0 8 2*SQRT(2) #18: --- - ----------- Simp(#17) 3 3 User #19: " => sqrt(2) [a modification of a problem due to W. Kahan] " - pi/4 / SQRT(2 - 2*COS(2*x)) #20: | ---------------------- dx User / 2 - 3*pi/4 #21: SQRT(2) Simp(#20) User #22: " Contour integrals => pi/a e^(-a) for a > 0. See Norman Levinson and" User #23: " Raymond M. Redheffer, _Complex Variables_, Holden-Day, Inc., 1970, p. 198. " #24: a :epsilon Real (0, inf) User #25: a Simp(#24) inf / COS(x) | --------- dx #26: | 2 2 User / x + a -inf inf / COS(x) 2*| --------- dx #27: | 2 2 Simp(#26) / x + a 0 User #28: " Integrand with a branch point => pi/sin(pi a) for 0 < a < 1 " #29: " [Levinson and Redheffer, p. 212] " User #30: a :epsilon Real (0, 1) User #31: a Simp(#30) inf / a - 1 | t #32: | -------- dt User / 1 + t 0 inf / a - 1 | t #33: | -------- dt Simp(#32) / t + 1 0 #34: a := User User #35: " Integrand with a residue at infinity => -2 pi [sin(pi/5) + sin(2 pi/5)] " #36: " (principal value) [Levinson and Redheffer, p. 234] "User inf / 3 | 5*x #37: | ---------------------- dx User | 2 3 4 / 1 + x + x + x + x -inf #38: ? Simp(#37) User #39: " integrate(1/[1 + x + x^2 + ... + x^(2 n)], x = -infinity..infinity) " User #40: " = 2 pi/(2 n + 1) [1 + cos(pi/[2 n + 1])] csc(2 pi/[2 n + 1]) " User #41: " [Levinson and Redheffer, p. 255] => 2 pi/5 [1 + cos(pi/5)] csc(2 pi/5) " inf / 1 | ----------------- dx #42: | 2 4 User / 1 + x + x + x -inf inf / 1 | ----------------- dx #43: | 4 2 Simp(#42) / x + x + x + 1 -inf User #44: " Integrand with a residue at infinity and a branch cut => pi [sqrt(2) - 1] " #45: " [Levinson and Redheffer, p. 234] " User 1 / 2 | SQRT(1 - x ) #46: | -------------- dx User | 2 / 1 + x -1 #47: pi*(SQRT(2) - 1) Simp(#46) User #48: " This is a common integral in many physics calculations " User #49: " => q/p sqrt(pi/p) e^(q^2/p) (Re p > 0) [Gradshteyn and Ryzhik 3.462(6)] " #50: p :epsilon (0, inf) User #51: p Simp(#50) inf / 2 #52: / x*EXP(- p*x + 2*q*x) dx User -inf 2 q /p SQRT(pi)*q*#e #53: ------------------- Simp(#52) 3/2 p #54: p := User User #55: " => 2 Euler's_constant [Gradshteyn and Ryzhik 8.367(5-6)] " 1 / / 1 1 / / 1 \\\ #56: | |-------- + ------- - LOG|LOG|---||| dt User / \ LOG(t) 1 - t \ \ t /// 0 1 1 / / 1 1 \ / #57: | |------- - -------| dt - / LN(- LN(t)) dt Simp(#56) / \ LN(t) t - 1 / 0 0 User #58: " This integral comes from atomic collision theory => 0 [John Prentice] " inf / SIN(t) #59: | --------*EXP(2*#i*t) dt User / t -inf #60: ? Simp(#59) #61: " => 1/12 [Gradshteyn and Ryzhik 6.443(3)] " User 1 / #62: / LOG(GAMMA(x))*COS(6*pi*x) dx User 0 1 / #63: / COS(6*pi*x)*LN((x - 1)!) dx Simp(#62) 0 #64: " => 36/35 [Gradshteyn and Ryzhik 7.222(2)] " User 1 / 3 #65: / (1 + x) *LEGENDRE_P(1, x)*LEGENDRE_P(2, x) dx User -1 36 #66: ---- Simp(#65) 35 #67: " => 1/sqrt(a^2 + b^2) (a > 0 and b real) " User #68: " [Gradshteyn and Ryzhik 6.611(1)] " User #69: a :epsilon Real (0, inf) User #70: a Simp(#69) #71: b :epsilon Real User #72: b Simp(#71) inf / #73: / EXP(- a*x)*BESSEL_J(0, b*x) dx User 0 Memory Full inf / #74: / EXP(- a*x)*JN(0, b*x) dx User 0 Memory Full #75: a := User #76: b := User User #77: " Integrand contains a special function => 4/(3 pi) [Tom Hagstrom] " inf / / BESSEL_J(1, x) \2 #78: | |----------------| dx User / \ x / 0 Memory Full inf / / JN(1, x) \2 #79: | |----------| dx User / \ x / 0 Memory Full #80: " => (cos 7 - 1)/7 [Gradshteyn and Ryzhik 6.782(3)] "User inf / #81: / CI(x)*BESSEL_J(0, 2*SQRT(7*x)) dx User 0 Memory Full inf / #82: / CI(x)*JN(0, 2*SQRT(7*x)) dx User 0 Memory Full User #83: " This integral comes from doing a two loop Feynman diagram for a QCD problem " User #84: " => - [17/3 + pi^2]/36 + log 2/9 [35/3 - pi^2/2 - 4 log 2 + log(2)^2] " #85: " + zeta(3)/4 = 0.210883... [Rolf Mertig] " User 1 / 2 / 1 \ #86: | x *POLYLOG|3, -------, inf| dx User / \ x + 1 / 0 1 / -k | 2 inf (x + 1) #87: | x *SUM ----------- dx Simp(#86) | k=1 3 / k 0 #88: Precision := Approximate User / 1 \ |/ 2 / 1 \ | #89: APPROX|| x *POLYLOG|3, -------, inf| dx| User |/ \ x + 1 / | \ 0 / #90: 0 Simp(#89) / 1 \ |/ 2 / 1 \ | #91: APPROX|| x *POLYLOG|3, -------, 15| dx| User |/ \ x + 1 / | \ 0 / #92: 0.210882 Simp(#91) User / 17 2 | ---- + pi / 2 #93: | 3 LOG(2) | 35 pi APPROX|- ------------ + --------*|---- - ----- - 4*LOG(2) + \ 36 9 \ 3 2 \ \ | 2| ZETA(3) | LOG(2) | + ---------| / 4 / #94: 0.210882 Simp(#93) #95: Precision := Exact User User #96: " Integrate a piecewise defined step function s(t) multiplied by cos t, where " User #97: " s(t) = 0 (t < 1); 1 (1 <= t <= 2); 0 (t > 2) " User #98: " => 0 (u < 1); sin u - sin 1 (1 <= u <= 2); sin 2 - sin 1 (u > 2) " #99: S(t) := IF(1 <= t AND t <= 2, 1, 0) User u / #100:/ S(t)*COS(t) dt User 0 u / #101:/ COS(t)*IF(1 <= t AND t <= 2, 1, 0) dt Simp(#100) 0 #102:S(t) := CHI(1, t, 2) User u / #103:/ S(t)*COS(t) dt User 0 Simp(#103) / SIN(2) SIN(u) \ / SIN(u) #104:SIGN(u - 2)*|-------- - --------| + SIGN(u - 1)*|-------- - \ 2 2 / \ 2 SIN(1) \ SIN(2) SIN(1) --------| + -------- - -------- 2 / 2 2 #105:s := User User #106:" Integrating first with respect to y and then x is much easier than " #107:" integrating first with respect to x and then y " User #108:" => (|b| - |a|) pi [W. Kahan] " User b / inf | / x | | --------- dy dx #109:| | 2 2 User | / x + y / -inf a #110:pi*|b| - pi*|a| Simp(#109) inf / b | / x | | --------- dx dy #111:| | 2 2 User | / x + y / a -inf #112:pi*|b| - pi*|a| Simp(#111) User #113:" => [log(sqrt(2) + 1) + sqrt(2)]/3 [Caviness et all, section 2.10.1] " 1 / 1 | / 2 2 #114:| / SQRT(x + y ) dx dy User / 0 0 LN(SQRT(2) + 1) SQRT(2) #115:----------------- + --------- Simp(#114) 3 3 #116:" => (pi a)/2 [Gradshteyn and Ryzhik 4.621(1)] " User pi/2 / pi/2 | / SIN(a)*SIN(y) | | ----------------------------------- dx dy #117:| | 2 2 2 User | / SQRT(1 - SIN(a) *SIN(x) *SIN(y) ) / 0 0 Memory Full #118:" => 46/15 [Paul Zimmermann] " User 1 / 2 | / | 2| #119:| / |y - x | dy dx User / 0 -1 46 #120:---- Simp(#119) 15 User #121:" Multiple integrals: volume of a tetrahedron => a b c / 6 " a / b*(1 - x/a) | / c*(1 - x/a - y/b) | | / #122:| | / 1 dz dy dx User | / 0 / 0 0 a*b*c #123:------- Simp(#122) 6 #124:" ---------- Quit ---------- " User