TSIL_COMPLEX TSIL_Bp (TSIL_REAL X, TSIL_REAL Y, TSIL_COMPLEX S, TSIL_REAL QQ)
{
if (X < TSIL_TOL) {
TSIL_Warn("Bp", "B(x',y) is undefined for x=0.");
return TSIL_Infinity;
}
if (TSIL_CABS(1.0L - S/(X+Y+2.0L*TSIL_SQRT(X*Y))) < TSIL_TOL) {
TSIL_Warn("Bp", "B(x',y) is undefined at s = (sqrt(x) + sqrt(y))^2.");
return TSIL_Infinity;
}
if (TSIL_CABS(S) < TSIL_TOL) {
if (TSIL_FABS(1.0L - X/Y) < TSIL_TOL)
return (-0.5L/X);
else
return 1.0L/(Y-X) + Y*TSIL_LOG(X/Y)/((Y-X)*(Y-X));
}
if (TSIL_CABS(1.0L - (X + Y - 2.0L*TSIL_SQRT(X*Y))/S) < TSIL_TOL)
return (1.0L - TSIL_SQRT(Y/X) +0.5L*TSIL_LOG(Y/X))/(X + Y - 2.0L*TSIL_SQRT(X*Y));
else
return ((X-Y-S)*TSIL_B(X,Y,S,QQ) + (X+Y-S)*TSIL_LOG(X/QQ)
-2.0L*TSIL_A(Y,QQ) + 2.0L*(S-X))/TSIL_Delta(S,X,Y);
}
TSIL_COMPLEX TSIL_Tx0y (TSIL_REAL X, TSIL_REAL Y, TSIL_COMPLEX S,
TSIL_REAL QQ)
{
TSIL_COMPLEX sqDeltaSXY, tp, tm, log1mtp, log1mtm;
TSIL_REAL lnbarX, lnbarY;
if (X < TSIL_TOL) {
TSIL_Warn("TSIL_Tx0y", "T(x,0,y) is undefined for x = 0.");
return TSIL_Infinity;
}
if (TSIL_CABS(S) < TSIL_TOL) return -TSIL_I2p(X, 0.0L, Y, QQ);
if (Y < TSIL_TOL) return TSIL_Tx00(X, S, QQ);
S = TSIL_AddIeps(S);
sqDeltaSXY = TSIL_CSQRT(X*X + Y*Y + S*S - 2.0L*X*Y - 2.0L*X*S - 2.0L*Y*S);
lnbarX = TSIL_LOG(X/QQ);
lnbarY = TSIL_LOG(Y/QQ);
tp = (Y - X + S + sqDeltaSXY)/(2.0L * Y);
tm = (Y - X + S - sqDeltaSXY)/(2.0L * Y);
log1mtp = TSIL_CLOG(1.0L - tp);
log1mtm = TSIL_CLOG(1.0L - tm);
return (-TSIL_Dilog (tp) - TSIL_Dilog (tm) + (1.0L - Y/S)*log1mtp*log1mtm
+ sqDeltaSXY * (log1mtp - log1mtm) / (2.0L * S)
+ lnbarX*lnbarY - 0.5L*(lnbarY + 1.0L)*(lnbarY + 1.0L)
+ (3.0L*S + Y - X) *(lnbarY-lnbarX)/(2.0L * S));
}
TSIL_COMPLEX TSIL_A (TSIL_REAL x, TSIL_REAL qq)
{
if (TSIL_FABS(x) < TSIL_TOL)
return 0.0;
if (x > 0)
return (x * (TSIL_LOG(x/qq) - 1.));
return (x * (TSIL_LOG(-x/qq) - 1. + I*PI));
}
TSIL_COMPLEX TSIL_Ap (TSIL_REAL x, TSIL_REAL qq)
{
if (TSIL_FABS(x) < TSIL_TOL)
return 0.0;
if (x > 0)
return (TSIL_LOG(x/qq));
return (TSIL_LOG(-x/qq) + I*PI);
}
TSIL_COMPLEX TSIL_B0x (TSIL_REAL X, TSIL_COMPLEX S, TSIL_REAL QQ)
{
if (TSIL_FABS (X) < TSIL_TOL)
return TSIL_B00(S,QQ);
if (TSIL_CABS (S) < TSIL_TOL)
return (1.0L - TSIL_LOG (X/QQ));
if (TSIL_CABS (1.0L - S/X) < 10.0L*TSIL_TOL)
return 2.0L - TSIL_LOG(X/QQ);
S = TSIL_AddIeps(S);
return 2.0L + ((X - S)*TSIL_CLOG((X - S)/QQ) - X*TSIL_LOG(X/QQ))/S;
}
TSIL_COMPLEX TSIL_TyyxAtx (TSIL_REAL X, TSIL_REAL Y, TSIL_REAL QQ)
{
TSIL_REAL lnbarX, lnbarY;
if (X < TSIL_TOL) return -TSIL_I2p(Y,Y,0,QQ);
if (Y < TSIL_TOL) {
TSIL_Warn("TSIL_TyyxAtx", "T(y,y,x) with s = 0 is undefined for x = 0.");
return TSIL_Infinity;
}
lnbarX = TSIL_LOG(X/QQ);
lnbarY = TSIL_LOG(Y/QQ);
return -0.5L + (1.0L - Y/X)*(TSIL_Dilog(1.0 -X/Y) -Zeta2)
+ lnbarX - 2.0L*lnbarY + 0.5L*lnbarY*lnbarY;
}
TSIL_COMPLEX TSIL_Trilogseries (TSIL_COMPLEX z)
{
TSIL_REAL absz = TSIL_CABS (z);
TSIL_REAL logepsilon = TSIL_LOG (TSIL_TOL);
TSIL_REAL mlogabsz;
TSIL_COMPLEX sum = z;
TSIL_COMPLEX ztothek;
TSIL_COMPLEX term;
TSIL_COMPLEX kcubed;
int k, kmax;
mlogabsz = -TSIL_CLOG (absz);
/*
The following kmax is hopefully designed to give accuracy to within
e^logepsilon, with some safety margin built in. Not completely
tested, but it seems good enough for government work anyway.
*/
kmax = 5 + (int) (( 6.0 -logepsilon -3.0 * log(-logepsilon)
+ 3.0 * log (mlogabsz)) / mlogabsz);
for (k = kmax; k > 1; k--)
{
ztothek = TSIL_CPOW (z, k);
kcubed = k*k*k;
term = ztothek/kcubed;
sum += term;
}
return sum;
}
TSIL_COMPLEX TSIL_TxyyAtx (TSIL_REAL X, TSIL_REAL Y, TSIL_REAL QQ)
{
TSIL_REAL lnbarX, lnbarY;
if (X < TSIL_TOL) {
TSIL_Warn("TSIL_TxyyAtx", "T(x,y,y) is undefined for s = x = 0.");
return TSIL_Infinity;
}
if (Y < TSIL_TOL) return TSIL_Tx00(X,X,QQ);
lnbarX = TSIL_LOG(X/QQ);
lnbarY = TSIL_LOG(Y/QQ);
return (-0.5L + (Y/X-1.0L)*(TSIL_Dilog(1.0 -X/Y) -Zeta2)
+ lnbarX*(lnbarY - 1.0L) - 0.5L*lnbarY*lnbarY);
}
TSIL_COMPLEX TSIL_B (TSIL_REAL X, TSIL_REAL Y, TSIL_COMPLEX S, TSIL_REAL QQ)
{
TSIL_REAL temp;
TSIL_COMPLEX sqDeltaSXY, lnbarX, lnbarY;
if (TSIL_FABS (X) < TSIL_FABS (Y)) {temp = Y; Y = X; X = temp;}
if (TSIL_FABS (X) < TSIL_TOL)
return TSIL_B00(S,QQ);
if (TSIL_FABS (Y) < TSIL_TOL)
return TSIL_B0x(X,S,QQ);
if (TSIL_CABS (S) < TSIL_TOL) {
if (TSIL_FABS (1.0L - Y/X) > 0.0L)
return (1.0L + (Y*TSIL_LOG(Y/QQ) - X*TSIL_LOG(X/QQ))/(X-Y));
else
return (-TSIL_LOG (X/QQ));
}
S = TSIL_AddIeps(S);
sqDeltaSXY = TSIL_CSQRT(TSIL_Delta(S, X, Y));
lnbarX = TSIL_LOG (X/QQ);
lnbarY = TSIL_LOG (Y/QQ);
/* Following avoids roundoff error for very negative s. */
if ((TSIL_CREAL(S) < -10.0L*(X+Y)) && (TSIL_CIMAG(S) < TSIL_TOL)) {
return (2.0L - 0.5L * (lnbarX + lnbarY) +
(sqDeltaSXY * TSIL_CLOG(0.5L*(X + Y - S + sqDeltaSXY)/Y) +
0.5L * (Y - X - sqDeltaSXY) * (lnbarX - lnbarY))/S);
}
return (2.0L - 0.5L * (lnbarX + lnbarY) +
(-sqDeltaSXY * TSIL_CLOG(0.5L*(X + Y - S - sqDeltaSXY)/X) +
0.5L * (Y - X - sqDeltaSXY) * (lnbarX - lnbarY))/S);
}
TSIL_COMPLEX TSIL_Tx00 (TSIL_REAL X, TSIL_COMPLEX S, TSIL_REAL QQ)
{
TSIL_REAL lnbarX;
if (X < TSIL_TOL) {
TSIL_Warn("TSIL_Tx00", "T(x,0,0) is undefined for x = 0.");
return TSIL_Infinity;
}
if (TSIL_CABS(S) < TSIL_TOL) return -TSIL_I2p(X, 0.0L, 0.0L, QQ);
lnbarX = TSIL_LOG(X/QQ);
if (TSIL_CABS (1.0L - S/X) < 10.0L*TSIL_TOL)
return 2.0L*Zeta2 -0.5L -lnbarX + 0.5L*lnbarX*lnbarX;
S = TSIL_AddIeps(S);
return Zeta2 - 0.5L + (1.0L - X/S)*TSIL_CLOG(1.0L -S/X)
-lnbarX + 0.5L*lnbarX*lnbarX + TSIL_Dilog (S/X);
}
