One-loop integrals at any precision

Scalar \(B_0\) and \(C_0\) in LoopTools conventions, to double precision and to 50 digits, and the Method-of-Brackets series for \(B_0\) compared with its closed form.

B0(5; 1, 1)          = (1.56959105903599596111056676705 + 1.40496294620814527863127492864j)
C0(0,0,10; 1, 2, 3)  = (-0.421932531674735453559401494473 - 0.336732268033680194357509591725j)
C0(0,0,10; 1, 2, 3)  = (-0.421932531674735453559401494473027777240303185 - 0.336732268033680194357509591724866396568161785j)
B0 closed form       = (1.5695910590359959611105667670493946 + 1.404962946208145278631274928640979j)
B0 MoB 2F1 series    = (1.5695910590359959611105667670493946 + 1.404962946208145278631274928640979j)
B0(5;1,2), mu2=91.19^2, delta=1/eps: (10.6675243794782430572441296241 + 0.0j)

import mpmath as mp

from ramanujan_qft import B0, B0_mob, C0

# Double precision: identical to LoopTools 2.16
print("B0(5; 1, 1)          =", B0(5, 1, 1))            # LoopTools: 1.5695910590 + 1.4049629462i
print("C0(0,0,10; 1, 2, 3)  =", C0(10, 1, 2, 3))         # LoopTools: -0.4219325317 - 0.3367322680i

# 50 digits, e.g. to validate another code or for cancellations in large expressions
print("C0(0,0,10; 1, 2, 3)  =", mp.nstr(C0(10, 1, 2, 3, dps=50), 45))

# The Method-of-Brackets series gives the same B0, continued across threshold:
print("B0 closed form       =", mp.nstr(B0(5, 1, 1, dps=40), 35))
print("B0 MoB 2F1 series    =", mp.nstr(B0_mob(5, 1, dps=40), 35))

# UV pole and scale: B0 = delta + log(mu^2) + ...
print("B0(5;1,2), mu2=91.19^2, delta=1/eps:", B0(5, 1, 2, mu2=91.19**2, delta=1))