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/* Author: Bill Gosper */
/*
Initial version generated from bffac.lisp with the commands
load("bffac.lisp");
bffac_functions:[OBFAC,AZETB,VONSCHTOONK,DIVRLST,BURN,OBZETA,BZETA,
BFPSI,BFFAC,BFHZETA,CBFFAC,BFPSI0,BFZETA];
apply(stringout,cons("bffac.mac",map('fundef,bffac_functions))),grind:true;
*/
prodhack_product (expr, i%, i%0, i%1) :=
(expr: subst ('i%, i%, expr),
if i%0 > i%1 then
subst (i%, 'i%, 1 / product (expr, i%, i%1 + 1, i%0 - 1))
else
subst (i%, 'i%, product (expr, i%, i%0, i%1)));
obfac(z,fpprec):=block([y:z-entier(z),x:entier(7*fpprec/3),m:0],
for k from entier(26*fpprec/3) step -1 thru 1 do m:x*(1/(y+k)-m/k),
bfloat(m)*x^y*prodhack_product(z-i+1,i,1,entier(z)));
azetb(s,fpp):=block([fpprec:15],
if s > 0
then block([m:0,
p:print(entier(log(
bfloat(2*%pi)^s
*bfloat(10)^(fpp:entier(fpp))
/bfloat(%pi*(s-1)!))
/bfloat(2*%pi)))],fpprec:fpp,
for k from entier(ev(%pi*p,numer)) step -1 thru 0 do
m:(s+2*k)*(s+2*k-1)*m/((2*k+1)*(2*k+2)*p^2)
+p*bern(2*k)/(s-1),(1/2+m)/p^s+sum(k^-s,k,1,p-1),
bfloat(%\%))
else (if s = 0 then -1/2
else ((2*%pi)^s*sin(%pi*s/2)*bffac(-s,fpprec)
*bzeta(1-s,fpprec)
/%pi,bfloat(%\%))));
vonschtoonk(p):=apply("*",
subst(["*" = lambda([[l]],1),"^" = lambda([[l]],1)],
factor(1+divrlst(p))));
divrlst(p):=(
args(expand(ratsimp(subst("^" = lambda([a,b],
(1-part(a)^(b/2+1))
/(1-part(a))),factor(p^2))))),
ev(%\%,part));
burn(p):=if evenp(p)
then block([d:vonschtoonk(p)],
block([fpprec:entier(ev(
(log(8*%pi*p)/2+p*(log(p/(2*%pi))-1)
+2*log(d))
/log(10),numer))
+1],
block([pi:bfloat(%pi)],
entier(2*d*p!*bfzeta(p,fpprec)/(2^p*pi^p)+1/2)
/d)))
*(-1)^(p/2-1) else bern(p);
obzeta(s,fpprec):=if s > 0
then block([m:0,
p:entier(ev((log(2*%pi)*s+log(10)*fpprec-log(%pi*(s-1)!))
/(2*%pi),numer))],
for k from entier(ev(%pi*p,numer)) step -1 thru 0 do
m:(s+2*k)*(s+2*k-1)*m/((2*k+1)*(2*k+2)*p^2)
+p*bern(2*k)/(s-1),(1/2+m)/p^s+sum(k^-s,k,1,p-1),
bfloat(%\%))
else (if s = 0 then -1/2
else ((2*%pi)^s*sin(%pi*s/2)*bffac(-s,fpprec)
*bzeta(1-s,fpprec)
/%pi,bfloat(%\%)));
bzeta(s,fpp):=if s > 0
then block([fpprec:7,m:0,t:0.0b0,p,
k:2*max(1,
entier(ev(3/2-(s-1/2)*log(s/(2*%pi))+log(10)*fpp
-log(%pi),numer)
/2))],
p:1+entier(bfloat((2*10^fpp*product(s+k-i-2,i,0,k-2)
/(2*3.14159265b0)^k)
^(1.0b0/(s+k-1)))),fpprec:fpp,
for k from k/2 step -1 thru 0 do
m:(s+2*k)*(s+2*k-1)*m/((2*k+1)*(2*k+2)*p^2)
+p*bern(2*k)/(s-1),
if s < 25 then t:sum(k^-s,k,1,p-1)
else for k from p-1 step -1 thru 1 do
(fpprec:fpp-entier(ev(s*log(k)/log(10),numer)),
t:t+k^-s),bfloat((1/2+m)/p^s+t))
else (if s = 0 then -1/2
else ((2*%pi)^s*sin(%pi*s/2)*bffac(-s,fpprec)
*bzeta(1-s,fpprec)
/%pi,bfloat(%\%)));
bfpsi(n,z,fpprec):=if equal(n,0) then bfpsi0(z,fpprec)
else bfhzeta(n+1,z,fpprec)*(-1)^(n-1)*n!;
bffac(z,fpprec):=if z < 0 then bfloat(%pi*z/sin(%pi*z))/bffac(-z,fpprec)
else block([k:2*(entier(0.41*fpprec)+1)],
block([m:1,y:(z+k)^2,x:0.0b0],
for i thru k/2 do
(m:m*(z+2*i-1)*(z+2*i),
x:(x+bern(k-2*i+2)/((k-2*i+1)*(k-2*i+2)))/y),
bfloat(sqrt(2)*sqrt(%pi)*sqrt(z+k)
*%e^((z+k)*(log(z+k)+x-1)))
/m));
bfhzeta(s,h,fpprec):=if s > 0
then block([m:0,p,
q:entier(ev((-log(%pi*(s-1)!)+log(2*%pi)*s
+log(10)*fpprec)
/(2*%pi)
-h+1,numer))
+h],p:max(q,h+1),
for k from entier(ev(%pi*q,numer)) step -1 thru 0 do
m:m*(s+2*k-1)*(s+2*k)/((2*k+1)*(2*k+2)*(p-1)^2)
+bern(2*k)*(p-1)/(s-1),
(m+1/2)/(p-1)^s+sum(1/(k+h)^s,k,0,entier(p-h-1.9b0)),
bfloat(%\%))
else (if s = 0 then 1/2-h else funmake('bfhzeta,[s,h,fpprec]));
cbffac(z,fpprec):=rectform(
if realpart(z:rectform(z)) < 0
then bfloat(%pi*z/sin(%pi*z))/cbffac(-z,fpprec)
else block([k:2*(entier(0.41*fpprec)+1)],
block([m:1,y:rectform(1/(z+k)^2),x:0.0b0],
for i thru k/2 do
(m:expand(m*(z+2*i-1)*(z+2*i)),
x:expand(
(x+bern(k-2*i+2)/((k-2*i+1)*(k-2*i+2)))
*y)),
bfloat(sqrt(2)*sqrt(%pi)*sqrt(z+k)
*%e^((z+k)*(log(z+k)+x-1)))
/m)));
bfpsi0(z,fpprec):=if z < 0 then bfloat(%pi*cot(%pi*(-z)))+bfpsi0(1-z,fpprec)
else block([k:2*(entier(0.41*fpprec)+1)],
block([m:0,y:(z+k)^2,x:0.0b0],
for i thru k/2 do
(m:1/(z+2*i-1)+1/(z+2*i-2)+m,
x:(x+bern(k-2*i+2)/(k-2*i+2))/y),
bfloat(log(z+k)-1/(2*(z+k))-x-m)));
bfzeta(s,fpp):=if s > 0
then block([fpprec:7,m:0,t:0.0b0,p,
k:2*max(1,
entier(ev(-(s-1/2)*log(s/(2*%pi))
+log(10)*fpp-log(%pi)+3/2,numer)
/2))],
p:entier(bfloat(2^(1/(s+k-1))*10^(fpp/(s+k-1))
*product(s+k-i-2,i,0,k-2)
^(1/(s+k-1))
/6.2831853b0^(k/(s+k-1))))
+1,fpprec:fpp,
for k from k/2 step -1 thru 0 do
m:m*(s+2*k-1)*(s+2*k)/((2*k+1)*(2*k+2)*p^2)
+bern(2*k)*p/(s-1),
if s < 25 then t:sum(1/k^s,k,1,p-1)
else for k from p-1 step -1 thru 1 do
(fpprec:fpp
-entier(ev(log(k)*s/log(10),numer)),
t:t+1/k^s),bfloat(t+(m+1/2)/p^s))
else (if s = 0 then -1/2
else (bfzeta(1-s,fpprec)*%pi^(s-1)*2^s*bffac(-s,fpprec)
*sin(%pi*s/2),bfloat(%\%)));
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