Question

Which of the following function is not called the Euler’s integral of the first kind?

(a) \(\beta(m, n) = \int_0^1 x^{m-1} (1-x)^{n-1} dx (m>0, n>0) \)

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

(c) \(\beta(m, n) = \int_0^∞ \frac{y^{n+1}}{(1+y)^{m+n}}  dy \)

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

The question was asked in an internship interview.

My question is from Special Functions topic in division Special Functions – Gamma, Beta, Bessel and Legendre of Engineering Mathematics

Answer 1

The correct answer is (b) \(\beta(m, n) = \int_0^{π/2} (sin⁡θ)^{2m-1}  (cos⁡θ)^{2n-1} dθ \)

Easiest explanation: Euler’s integral of the first kind is nothing but Beta function. So, here only \( \beta(m, n) = \int_0^{π/2}(sin⁡θ)^{2m-1}  (cos⁡θ)^{2n-1} dθ\) is not the definition of Beta function.