porting some code for complex lgamma, beta, eta, and zeta functions

examples/specfun.j

@@ -0,0 +1,165 @@
+function angle_restrict_symm(theta)
+ P1 = 4 * 7.8539812564849853515625e-01
+ P2 = 4 * 3.7748947079307981766760e-08
+ P3 = 4 * 2.6951514290790594840552e-15
+
+ y = 2*floor(theta/(2*pi))
+ r = ((theta - y*P1) - y*P2) - y*P3
+ if (r > pi)
+ r -= (2*pi)
+ end
+ return r
+end
+
+const clg_coeff = [76.18009172947146,
+ -86.50532032941677,
+ 24.01409824083091,
+ -1.231739572450155,
+ 0.1208650973866179e-2,
+ -0.5395239384953e-5]
+
+function clgamma_lanczos(z)
+ sqrt2pi = 2.5066282746310005
+
+ y = x = z
+ temp = x + 5.5
+ zz = log(temp)
+ zz = zz * (x+0.5)
+ temp -= zz
+ ser = complex(1.000000000190015, 0)
+ for j=1:6
+ y += 1.0
+ zz = clg_coeff[j]/y
+ ser += zz
+ end
+ zz = sqrt2pi*ser / x
+ return log(zz) - temp
+end
+
+function lgamma(z::Complex)
+ if real(z) <= 0.5
+ a = clgamma_lanczos(1-z)
+ b = log(sin(pi * z))
+ logpi = 1.14472988584940017
+ z = logpi - b - a
+ else
+ z = clgamma_lanczos(z)
+ end
+ complex(real(z), angle_restrict_symm(imag(z)))
+end
+
+beta(x::Number, w::Number) = exp(lgamma(x)+lgamma(w)-lgamma(x+w))
+
+const eta_coeffs =
+ [.99999999999999999997,
+ -.99999999999999999821,
+ .99999999999999994183,
+ -.99999999999999875788,
+ .99999999999998040668,
+ -.99999999999975652196,
+ .99999999999751767484,
+ -.99999999997864739190,
+ .99999999984183784058,
+ -.99999999897537734890,
+ .99999999412319859549,
+ -.99999996986230482845,
+ .99999986068828287678,
+ -.99999941559419338151,
+ .99999776238757525623,
+ -.99999214148507363026,
+ .99997457616475604912,
+ -.99992394671207596228,
+ .99978893483826239739,
+ -.99945495809777621055,
+ .99868681159465798081,
+ -.99704078337369034566,
+ .99374872693175507536,
+ -.98759401271422391785,
+ .97682326283354439220,
+ -.95915923302922997013,
+ .93198380256105393618,
+ -.89273040299591077603,
+ .83945793215750220154,
+ -.77148960729470505477,
+ .68992761745934847866,
+ -.59784149990330073143,
+ .50000000000000000000,
+ -.40215850009669926857,
+ .31007238254065152134,
+ -.22851039270529494523,
+ .16054206784249779846,
+ -.10726959700408922397,
+ .68016197438946063823e-1,
+ -.40840766970770029873e-1,
+ .23176737166455607805e-1,
+ -.12405987285776082154e-1,
+ .62512730682449246388e-2,
+ -.29592166263096543401e-2,
+ .13131884053420191908e-2,
+ -.54504190222378945440e-3,
+ .21106516173760261250e-3,
+ -.76053287924037718971e-4,
+ .25423835243950883896e-4,
+ -.78585149263697370338e-5,
+ .22376124247437700378e-5,
+ -.58440580661848562719e-6,
+ .13931171712321674741e-6,
+ -.30137695171547022183e-7,
+ .58768014045093054654e-8,
+ -.10246226511017621219e-8,
+ .15816215942184366772e-9,
+ -.21352608103961806529e-10,
+ .24823251635643084345e-11,
+ -.24347803504257137241e-12,
+ .19593322190397666205e-13,
+ -.12421162189080181548e-14,
+ .58167446553847312884e-16,
+ -.17889335846010823161e-17,
+ .27105054312137610850e-19]
+
+function eta(z)
+ if z == 0
+ return complex(0.5)
+ end
+ re, im = reim(z)
+ if im==0 && re < 0 && integer_valued(re) && re==round(re/2)*2
+ return complex(0.0)
+ end
+ reflect = false
+ if re < 0.5
+ re = 1-re
+ im = -im
+ reflect = true
+ end
+ dn = float64(length(eta_coeffs))
+ sr = 0.0
+ si = 0.0
+ for n = length(eta_coeffs):-1:1
+ p = (dn^-re) * eta_coeffs[n]
+ lnn = -im * log(dn)
+ sr += p * cos(lnn)
+ si += p * sin(lnn)
+ dn -= 1
+ end
+ if reflect
+ z = complex(re, im)
+ b = 2.0 - 2.0^complex(re+1,im)
+
+ f = 2.0^z - 2
+ piz = pi^z
+
+ b = b/f/piz
+
+ return complex(sr,si) * exp(lgamma(z)) * b * cos(pi/2*z)
+ end
+ return complex(sr, si)
+end
+
+eta(x::Real) = real(eta(complex(float64(x))))
+
+function zeta(z::Complex)
+ zz = 2.0^z
+ eta(z) * zz/(zz-2)
+end
+
+zeta(x::Real) = real(zeta(complex(float64(x))))