Question

Which of the following is not the definition of Beta function?

(a) \(\beta(m, n) = 2\int_0^1 x^{m-1} (1-x)^{n-1} dx (m>0, n>0) \)

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

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

(d) \(\beta(m, n) = \int_0^1 \frac{x^{m-1}+x^{n-1}}{(1+x)^{m+n}}  dx \)

This question was posed to me in an interview for job.

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

Answer 1

The correct option is (a) \(\beta(m, n) = 2\int_0^1 x^{m-1} (1-x)^{n-1} dx (m>0, n>0) \)

Explanation: \(\beta(m, n)\) can be written as either \(\int_0^1 x^{m-1} (1-x)^{n-1} dx \, (m>0, n>0)\)

(or)

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

(or)

\( = \int_0^∞ \frac{y^{n+1}}{\left(1+y\right)^{m+n}}  dy \;or \int_0^1 \frac{x^{m-1}+x^{n-1}}{\left(1+x\right)^{m+n}}  dx\).

So the correct answer is \(2\int_0^1 x^{m-1} (1-x)^{n-1} dx (m>0, n>0) \) which is actually not the formula for Beta function.