We have a well-known identity $$B(a,b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$$ between beta function and gamma function.
I have a complicated expression involving the terms in the form $\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$. I want to simplify by replacing it to $B(a,b)$. How can I do this?
The following is the example that I want to simplify:
(2^(-3 + 4/m)
E^(-((3 I \[Pi])/m)) (-1 + E^((2 I \[Pi])/m))^3 Gamma[(-2 + m)/
m] Gamma[(-1 + m)/m] Gamma[
1/m + (I w3)/
2] (Beta[(-1 + m)/m, 1/(2 m) + 1/4 I (w1 + w3)] Gamma[
1/4 (4 - 2/m - 2 I w1 + I w3)] Gamma[(2 + 2 I m w1 - I m w3)/(
4 m)] - Beta[(-1 + m)/m, 1/(2 m) - 1/4 I (w1 + w3)] Gamma[
1/4 (4 - 2/m + 2 I w1 - I w3)] Gamma[(2 - 2 I m w1 + I m w3)/(
4 m)]))/(Gamma[1 - 1/m + (I w3)/2] Gamma[
1/4 (4 - 2/m + 2 I w1 - I w3)] Gamma[
1/4 (4 - 2/m - 2 I w1 + I w3)])
(Looking carefully, one can see that the above simplification is possible.)
B[(-1 + m)/m, 1/(2 m) - 1/4 I (w1 + w3)]
readBeta[(-1 + m)/m, 1/(2 m) - 1/4 I (w1 + w3)]
? $\endgroup$