#1: " ----------[ D e r i v e ]---------- " User #2: " ---------- Initialization ---------- " User #3: " ---------- Special Functions ---------- " User User #4: " Bernoulli numbers: B_16 => -3617/510 [Gradshteyn and Ryzhik 9.71] " #5: BERNOULLI(16) User 3617 #6: - ------ Simp(#5) 510 User #7: " d/dk E(phi, k) => [E(phi, k) - F(phi, k)]/k where F(phi, k) and E(phi, k) " User #8: " are elliptic integrals of the 1st and 2nd kind, respectively " #9: " [Gradshteyn and Ryzhik 8.123(3)] " User d 2 #10: -- ELLIPTIC_E(phi, k ) User dk phi / 2 | SIN(t_) #11: - k*| ----------------------- dt_ Simp(#10) | 2 2 / SQRT(1 - k *SIN(t_) ) 0 2 2 ELLIPTIC_E(phi, k ) - ELLIPTIC_F(phi, k ) #12: ------------------------------------------- User k Simp(#12) phi phi ~ / 2 2 / 1 ~ / SQRT(1 - k *SIN(t_) ) dt_ - | ----------------------- ~ 0 | 2 2 ~ #13: / SQRT(1 - k *SIN(t_) ) ~ 0 ~ ---------------------------------------------------------------~ k ~ dt_ ---- User #14: " Jacobian elliptic functions: d/du dn u => -k^2 sn u cn u " #15: " [Gradshteyn and Ryzhik 8.158(3)] " User #16: "DIF(dn(u), u)" User #17: " => -2 sqrt(pi) [Gradshteyn and Ryzhik 8.338(3)] " User / 1 \ #18: GAMMA|- ---| User \ 2 / #19: - 2*SQRT(pi) Simp(#18) User #20: " psi(1/3) => - Euler's_constant - pi/2 sqrt(1/3) - 3/2 log 3 where psi(x) " User #21: " is the psi function [= d/dx log Gamma(x)] [Gradshteyn and Ryzhik 8.366(6)] " / 1 \ #22: PSI|---| User \ 3 / Simp(#22) #23: - LN(3) - / inf inf ~ | / t_ / ~ 3*|6*| --------------------------- dt_ - 6*| ---------~ | | pi*t_ 2 | pi*t_~ | / (#e - 1)*(9*t_ + 1) / (#e ~ \ 0 0 ~ -------------------------------------------------------------~ 2 ~ \ t_ | ------------------ dt_ + 1| 2 | + 1)*(9*t_ + 1) | / ---------------------------- User #24: " Bessel function of the first kind of order 2 => 0.04158 + 0.24740 i " #25: Precision := Approximate User #26: APPROX(BESSEL_J(2, 1 + #i)) User #27: 0.0415798 + 0.247397*#i Simp(#26) #28: APPROX(JN(2, 1 + #i)) User #29: 0.0415799 + 0.247397*#i Simp(#28) #30: Precision := Exact User #31: " => 12/pi^2 [Gradshteyn and Ryzhik 8.464(6)] " User / 5 pi \ #32: BESSEL_J|- ---, ----| User \ 2 2 / #33: 0 Simp(#32) / 5 pi \ #34: JN|- ---, ----| User \ 2 2 / 12 ----- #35: 2 Simp(#34) pi User #36: " => sqrt(2/(pi z)) (sin z/z - cos z) [Gradshteyn and Ryzhik 8.464(3)] " / 3 \ #37: BESSEL_J|---, z| User \ 2 / Simp(#37) / / | | | / 3 \ | / 3 \ / 3 #38: IF|z = 0, IF|--- = 0, 1, 0|, IF|MOD|---| = 0, IF|--- < 0, | \ 2 / | \ 2 / \ 2 | | \ \ 3/2 \ / / / PrecisionDigits (-1) , 1|*|BESSEL_J_LIST|4 + FLOOR|----------------- + / \ \ \ 2 \ | 3 | \\ 4*|z|| + |---|, z|| , / | 2 | // ABS(3/2) + 1 / z \3/2 |---| pi \ 2 / / 2*(3/2) ---------------------------*/ COS(z*COS(t_))*SIN(t_) / 3 1 \ 0 SQRT(pi)*GAMMA|--- + ---| \ 2 2 / \\ || || dt_|| || || // / 3 \ #39: q_ := JN|---, z| User \ 2 / #40: q_ User SQRT(2)*SIN(z) SQRT(2)*COS(z) ---------------- - ------------------ #41: 3/2 SQRT(pi)*SQRT(z) Simp(#40) SQRT(pi)*z #42: FACTOR(q_) User SQRT(2)*(SIN(z) - z*COS(z)) ----------------------------- #43: 3/2 Simp(#42) SQRT(pi)*z User #44: " d/dz J_0(z) => - J_1(z) [Gradshteyn and Ryzhik 8.473(4)] " d #45: -- BESSEL_J(0, z) User dz d #46: -- JN(0, z) User dz 1 / alpha*SIN(alpha*z) 2*| -------------------- dalpha | 2 #47: / SQRT(1 - alpha ) Simp(#46) 0 - ---------------------------------- pi User #48: " Associated Legendre (spherical) function of the 1st kind: P^mu_nu(0) " User #49: " => 2^mu sqrt(pi) / [Gamma([nu - mu]/2 + 1) Gamma([- nu - mu + 1]/2)] " #50: " [Gradshteyn and Ryzhik 8.756(1)] " User #51: ASSOCIATED_LEGENDRE_P(nu, mu, 0) User mu /d \mu #52: (-1) *|--| LEGENDRE_P(nu, 0) Simp(#51) \d0/ #53: " P^1_3(x) => -3/2 sqrt(1 - x^2) (5 x^2 - 1) " User #54: " [Gradshteyn and Ryzhik 8.813(4)] " User #55: ASSOCIATED_LEGENDRE_P(3, 1, x) User 2 2 3*(1 - 5*x )*SQRT(1 - x ) #56: --------------------------- Simp(#55) 2 User #57: " nth Chebyshev polynomial of the 1st kind: T_n(x) => 0 " #58: " [Gradshteyn and Ryzhik 8.941(1)] " User User #59: CHEBYCHEV_T(1008, x) - 2*x*CHEBYCHEV_T(1007, x) + CHEBYCHEV_T(1006, x) #60: TN(1008, x) - 2*x*TN(1007, x) + TN(1006, x) User #61: " T_n(-1) => (-1)^n [Gradshteyn and Ryzhik 8.944(2)] "User #62: n :epsilon Integer User #63: n Simp(#62) #64: CHEBYCHEV_T(n, -1) User Simp(#64) / / / #65: IF|n = 0, 1, IF|n = 1, -1, IF|MOD(n, 2) = 0, \ \ \ / n \2 2*CHEBYCHEV_T_NUMERIC|---, -1| - 1, \ 2 / / n - 1 2*CHEBYCHEV_T_NUMERIC|-------, \ 2 \ / n + 1 \ \\\ -1|*CHEBYCHEV_T_NUMERIC|-------, -1| - -1||| / \ 2 / /// #66: TN(n, -1) User #67: TN_(n, -1) Simp(#66) #68: n := User #69: " => arcsin z/z [Gradshteyn and Ryzhik 9.121(26)] " User / 1 1 3 2\ #70: GAUSS|---, ---, ---, z | User \ 2 2 2 / Simp(#70) / 2 \ | SQRT(2 - z ) | #71: ATAN|--------------| pi \ z / ------- - ---------------------- + 2*|z| z / / 2 \\ 2 | 2 | z - 2 || #i*SIGN(z - 1)*LN||z| + SQRT(z - 1)*SQRT|--------|| | | 2 || \ \ z - 1 // ------------------------------------------------------- - |z| 2 2 #i*SIGN(z - 1)*LN(|z| + SQRT(z - 1)) ---------------------------------------- - |z| 2 #i*LN(2)*SIGN(z - 1) ----------------------- 2*|z| / 1 1 3 2\ #72: F21|---, ---, ---, z | User \ 2 2 2 / ASIN(z) #73: --------- Simp(#72) z User #74: " => sin(n z)/(n sin z cos z) [Gradshteyn and Ryzhik 9.121(17)] " #75: n :epsilon Integer User #76: n Simp(#75) / n + 2 n - 2 3 2\ #77: GAUSS|-------, - -------, ---, SIN(z) | User \ 2 2 2 / Simp(#77) / 1/2 ~ |/ n~ || SQRT(t)*SQRT(t ~ SQRT(pi)*|| ---------------------------------------------~ || n 2~ #78: |/ (t - 1)*SQRT((1 - t) )*SQRT(((1 - t)*COS(z) ~ \ 0 ~ - -------------------------------------------------------------~ ~ ~ ~ 1/2 ~ / ~ ) | S~ ------------------------------ dt + | --------------------~ n 2 | ~ + t) )*((t - 1)*COS(z) - t) / (t - 1)*SQRT((1 - t~ 0 ~ -------------------------------------------------------------~ 2 / n 1 \ / ~ 2*COS(z) *|--- - ---|!*|-~ \ 2 2 / \ ~ 1/2 ~ n / ~ QRT(t)*SQRT(t ) | ~ --------------------------------- dt - | -----------------~ n 2 n | ~ ) )*SQRT(((1 - t)*COS(z) + t) ) / SQRT(t)*SQRT((1 ~ 0 ~ -------------------------------------------------------------~ n \ ~ ---|! ~ 2 / ~ \ n | SQRT(t ) | ------------------------------------ dt| n 2 n | - t) )*SQRT(((1 - t)*COS(z) + t) ) | / ----------------------------------------- - 1/2 ~ / n ~ | SQRT((1 - t) ) ~ SQRT(pi)*| ----------------------------------------------~ | n 2 n ~ / SQRT(1 - t)*SQRT(t )*SQRT((1 - t*SIN(z) ) )*(~ 0 ~ -------------------------------------------------------------~ / n 1 \ / n \ ~ 2*|--- - ---|!*|- ---|! ~ \ 2 2 / \ 2 / ~ --------------- dt 2 t*SIN(z) - 1) ------------------- / n + 2 n - 2 3 2\ #79: q_ := F21|-------, - -------, ---, SIN(z) | User \ 2 2 2 / #80: q_ User Simp(#80) / | #81: (n - 2)*FLOOR(z/pi + 1/2) | 2*COS(z*(n - 2)) (-1) *|------------------ + \ n / 1 2*SIN(z) \ \ SIN(z*(n - 2))*|---------- - ----------| | \ n*SIN(z) n / | ------------------------------------------| COS(z) / #82: Trigonometry := Expand User #83: q_ User n*FLOOR(z/pi + 1/2) (-1) *SIN(n*z) #84: ---------------------------------- Simp(#83) n*SIN(z)*COS(z) #85: Trigonometry := Auto User #86: n := User User #87: " zeta'(0) => - 1/2 log(2 pi) [Gradshteyn and Ryzhik 9.542(4)] " d #88: lim -- ZETA(x) User x->0 dx d #89: lim -- ZETA(x) Simp(#88) x->0 dx #90: " Dirac delta distribution => 3 f(4/5) + g'(1) " User User #91: "INT(f((x + 2)/5)*delta((x - 2)/3) - g(x)*DIF(delta(x - 1), x), x, 0, 3)" #92: " Define an antisymmetric function f " User #93: " Test it out => [-f(a, b, c), 0] " User #94: F(x, y, z) := User #95: [F(c, b, a), F(c, b, c)] User #96: [F(c, b, a), F(c, b, c)] Simp(#95) #97: f := User #98: " ---------- Quit ---------- " User