Question

What is the value of \(\beta(m, \frac{1}{2}) \)?

(a) β(m, m)

(b) 2^2m-1 β(m, m)

(c) 2^2m+1 β(m, m)

(d) 2^2m β(m, m)

I had been asked this question in final exam.

My question comes from Special Functions in chapter Special Functions – Gamma, Beta, Bessel and Legendre of Engineering Mathematics

Answer 1

Correct answer is (b) 2^2m-1 β(m, m)

Easy explanation: \(\beta(m, \frac{1}{2}) =  2\int_0^{π/2}(sin⁡θ)^{2m-1}   dθ \)

\(\beta(m, m) = 2 \int_0^{π/2}(sin⁡θ)^{2m-1}  (cos⁡θ)^{2m-1} dθ \)

\( = 2^{-2m+2} \int_0^{π/2}(2 sinθ cos⁡θ)^{2m-1} dθ \)

Substituting 2θ=φ,

\( = 2^{-2m+1} \int_0^π sin⁡φdφ \)

\( = 2^{-2m+1}.2.\int_0^{π/2}sin⁡φdφ \)

\( = \frac{1}{2^{2m-1}}  \beta(m, \frac{1}{2}). \)