Special Functions

The study material of Special Functions are given on this page.

Special Functions Practice Problems. Some practice problems of Special Functions are given here.

Mid Term Practice Problems

1. What is the value of $\Gamma(7)$?
2. Find the Value of $B\left(\dfrac{1}{2}, \dfrac{1}{2} \right)\quad$ [Solution]
3. Find the Value of $\Gamma\left(\dfrac{1}{2}\right)\quad$ [Solution]
4. For what values of m and n, the integral $\displaystyle \int_0^1 x^{m+1}(1-x)^{1-n}\,dx \quad$ converges.
5. $\displaystyle \int_0^\infty e^{-x^2}\,dx \quad$ [Solution]
6. $\displaystyle \int_{-\infty}^\infty e^{-x^2}\,dx \quad$ [Solution]
7. Evaluate $\displaystyle \int_0^{\frac{\pi}{2}} \sqrt{\tan x}\,dx$.
8. Using the beta-gamma functions, find $\displaystyle \int_0^{\frac{\pi}{2}} \sin^3x \cos^4x\,dx$.
9. Define an orthogonal sequence. Prove that the sequence $\{\cos n\theta\}_{n=1}^\infty$ is an orthogonal sequence over the interval $(0, \pi)$.
10. Show that the sequence $\{T_n(x)\}_{n=1}^\infty$ where T_n(x) = cos nθ = cos(n cos^-1x), -1 ≤ x ≤ 1, is an orthogonal polynomial sequence over (0, 1) with respect to the weight $(1-x^2)^{-1/2}$.

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