Sun Apr 5 13:10:03 MDT 1998 euler% math Mathematica 3.0 for Solaris Copyright 1988-96 Wolfram Research, Inc. -- Terminal graphics initialized -- In[1]:= In[2]:= In[3]:= (* ----------[ M a t h e m a t i c a ]---------- *) 0. Second In[4]:= (* ---------- Initialization ---------- *) 0. Second In[5]:= (* ---------- Special Functions ---------- *) 0. Second In[6]:= (* Bernoulli numbers: B_16 => -3617/510 [Gradshteyn and Ryzhik\ > 9.71] *) 0. Second In[7]:= BernoulliB[16] 0.01 Second 3617 Out[7]= -(----) 510 In[8]:= (* d/dk E(phi, k) => [E(phi, k) - F(phi, k)]/k where F(phi, k) and\ > E(phi, k) are elliptic integrals of the 1st and 2nd kind, respectively [Gradshteyn and Ryzhik 8.123(3)] *) 0. Second In[9]:= D[EllipticE[phi, k^2], k] 0. Second 2 2 EllipticE[phi, k ] - EllipticF[phi, k ] Out[9]= --------------------------------------- k In[10]:= (* Jacobian elliptic functions: d/du dn u => -k^2 sn u cn u [Gradshteyn and Ryzhik 8.158(3)] *) 0. Second In[11]:= D[JacobiDN[u], u] 0.13 Second Out[11]= JacobiDN'[u] In[12]:= (* => -2 sqrt(pi) [Gradshteyn and Ryzhik 8.338(3)] *) 0. Second In[13]:= Gamma[-1/2] 0. Second Out[13]= -2 Sqrt[Pi] In[14]:= (* psi(1/3) => - Euler's_constant - pi/2 sqrt(1/3) - 3/2 log 3 \ > where psi(x) is the psi function [= d/dx log Gamma(x)] [Gradshteyn and Ryzhik\ > 8.366(6)] *) 0. Second In[15]:= PolyGamma[1/3] 0. Second 1 Out[15]= PolyGamma[0, -] 3 In[16]:= (* Bessel function of the first kind of order 2 => 0.04158 + 0.24740\ > i *) 0. Second In[17]:= N[BesselJ[2, 1 + I]] 0. Second Out[17]= 0.0415799 + 0.247398 I In[18]:= (* => 12/pi^2 [Gradshteyn and Ryzhik 8.464(6)] *) 0. Second In[19]:= BesselJ[-5/2, Pi/2] 0. Second 5 Pi Out[19]= BesselJ[-(-), --] 2 2 In[20]:= FullSimplify[%] 0.38 Second 12 Out[20]= --- 2 Pi In[21]:= (* => sqrt(2/(pi z)) (sin z/z - cos z) [Gradshteyn and Ryzhik\ > 8.464(3)] *) 0. Second In[22]:= BesselJ[3/2, z] 0.01 Second 2 Sin[z] Sqrt[--] (-Cos[z] + ------) Pi z Out[22]= --------------------------- Sqrt[z] In[23]:= (* d/dz J_0(z) => - J_1(z) [Gradshteyn and Ryzhik 8.473(4)] *) 0. Second In[24]:= D[BesselJ[0, z], z] 0. Second BesselJ[-1, z] - BesselJ[1, z] Out[24]= ------------------------------ 2 In[25]:= FullSimplify[%] 0.07 Second Out[25]= -BesselJ[1, z] In[26]:= (* Associated Legendre (spherical) function of the 1st kind:\ > P^mu_nu(0) => 2^mu sqrt(pi) / [Gamma([nu - mu]/2 + 1) Gamma([- nu - mu + 1]/2)] [Gradshteyn and Ryzhik 8.756(1)] *) 0. Second In[27]:= LegendreP[nu, mu, 0] 0. Second Out[27]= LegendreP[nu, mu, 0] In[28]:= (* P^1_3(x) => -3/2 sqrt(1 - x^2) (5 x^2 - 1) [Gradshteyn and Ryzhik 8.813(4)] *) 0. Second In[29]:= LegendreP[3, 1, x] 0.02 Second 2 2 -3 Sqrt[1 - x ] (-1 + 5 x ) Out[29]= --------------------------- 2 In[30]:= (* nth Chebyshev polynomial of the 1st kind: T_n(x) => 0 [Gradshteyn and Ryzhik 8.941(1)] *) 0. Second In[31]:= Simplify[ChebyshevT[1008, x] - 2*x*ChebyshevT[1007, x] +\ > ChebyshevT[1006, x]] 13.01 Second Out[31]= 0 In[32]:= (* T_n(-1) => (-1)^n [Gradshteyn and Ryzhik 8.944(2)] *) 0. Second In[33]:= ChebyshevT[n, -1] 0. Second Out[33]= ChebyshevT[n, -1] In[34]:= FullSimplify[%] 0.09 Second Out[34]= Cos[n Pi] In[35]:= (* => arcsin z/z [Gradshteyn and Ryzhik 9.121(26)] *) 0. Second In[36]:= HypergeometricPFQ[{1/2, 1/2}, {3/2}, z^2] 0.18 Second 2 ArcSin[Sqrt[z ]] Out[36]= ---------------- 2 Sqrt[z ] In[37]:= PowerExpand[%] 0. Second ArcSin[z] Out[37]= --------- z In[38]:= Hypergeometric2F1[1/2, 1/2, 3/2, z^2] 0.12 Second 2 ArcSin[Sqrt[z ]] Out[38]= ---------------- 2 Sqrt[z ] In[39]:= (* => sin(n z)/(n sin z cos z) [Gradshteyn and Ryzhik 9.121(17)] *) 0. Second In[40]:= HypergeometricPFQ[{(n + 2)/2, -(n - 2)/2}, {3/2}, Sin[z]^2] 0.04 Second 1 n 2 Sin[(-1 + 2 (- + -)) ArcSin[Sqrt[Sin[z] ]]] 2 2 Out[40]= ------------------------------------------- 2 2 n Sqrt[Sin[z] ] Sqrt[1 - Sin[z] ] In[41]:= Simplify[%] 0.52 Second 2 2 Sin[n ArcSin[Sqrt[Sin[z] ]]] Out[41]= ------------------------------ 2 n Sqrt[Sin[2 z] ] In[42]:= PowerExpand[%] 0.02 Second 2 Csc[2 z] Sin[n z] Out[42]= ------------------- n In[43]:= Hypergeometric2F1[(n + 2)/2, -(n - 2)/2, 3/2, Sin[z]^2] 0.03 Second 1 n 2 Sin[(-1 + 2 (- + -)) ArcSin[Sqrt[Sin[z] ]]] 2 2 Out[43]= ------------------------------------------- 2 2 n Sqrt[Sin[z] ] Sqrt[1 - Sin[z] ] In[44]:= (* zeta'(0) => - 1/2 log(2 pi) [Gradshteyn and Ryzhik 9.542(4)] *) 0. Second In[45]:= D[Zeta[x], x] /. x -> 0 0. Second -Log[2 Pi] Out[45]= ---------- 2 In[46]:= (* Dirac delta distribution => 3 f(4/5) + g'(1) *) 0. Second In[47]:= << Calculus`DiracDelta` 0.46 Second In[48]:= Integrate[f[(x + 2)/5]*DiracDelta[(x - 2)/3] - g[x]*DiracDelta'[x -\ > 1], {x, 0, 3}] 0.65 Second 4 Out[48]= 3 f[-] + g'[1] 5 In[49]:= (* Define an antisymmetric function f *) 0. Second In[50]:= f[l__]:= Signature[{l}]*Apply[HoldForm[f], Sort[{l}]] 0. Second In[51]:= (* Test it out => [-f(a, b, c), 0] *) 0. Second In[52]:= {f[c, b, a], f[c, b, c]} 0. Second Out[52]= {-f[a, b, c], 0} In[53]:= Clear[f] 0. Second In[54]:= (* ---------- Quit ---------- *) 0. Second In[55]:= Quit[] real 19.56 user 16.66 sys 0.30