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. Define beta and gamma functions.
2. What is the value of $\Gamma(7)$?
3. Prove that $\Gamma(n+1)=n!$
4. Find the Value of $B\left(\dfrac{1}{2}, \dfrac{1}{2} \right)\quad$ [Solution]
5. Find the Values of $\Gamma\left(\dfrac{1}{2}\right)$ and $\Gamma\left(\dfrac{5}{2} \right)\quad$ [Solution]
6. For what values of m and n, the integral $\displaystyle \int_0^1 x^{m+1}(1-x)^{1-n}\,dx \quad$ converges.
7. $\displaystyle \int_0^\infty e^{-x^2}\,dx \quad$ [Solution]
8. $\displaystyle \int_{-\infty}^\infty e^{-x^2}\,dx \quad$ [Solution]
9. Using the beta-gamma functions, find $\displaystyle \int_0^{\frac{\pi}{2}} \sin^2 \theta \cos^3\theta\,d\theta$.
10. Show that $\Gamma(n)=\displaystyle \int_0^\infty e^{-y^{\frac{1}{n}}} \,dy$.
11. Define an orthogonal sequence. Prove that the sequence $\{\cos n\theta\}_{n=1}^\infty$ is an orthogonal sequence over the interval $(0, \pi)$.
12. 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 $(1-x^2)^{-1/2}$.
13. Find the generating function for the Hermite polynomials 𝐻_n(𝑥). Compute 𝐻_2n(0) and 𝐻_2n+1(0).
14. Find the generating function for Charlier polynomials. Also, find its orthogonality relation.

Study Material:

Charlier Polynomials

Hermite Polynomials

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