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 2026

Q1: Define beta and gamma functions.

Q2: What are the value of $\Gamma(6)$ and $B(4, 5)$?

Q3: Prove that $\Gamma(n+1)=n \Gamma(n)$, and hence prove $\Gamma(n+1)=n!$

Q4: Find the Value of $B\left(\dfrac{1}{2}, \dfrac{1}{2} \right)\quad$ [Solution]

Q5: Find the Values of $\Gamma\left(\dfrac{1}{2}\right)$ and $\Gamma\left(\dfrac{5}{2} \right)\quad$ [Solution]

Q6: For what values of m and n, the integral $\displaystyle \int_0^1 x^{2m+1}(1-x)^{3n-1}\,dx \quad$ converges.

Q7: $\displaystyle \int_0^\infty e^{-x^2}\,dx \quad$ [Solution]

Q8: $\displaystyle \int_{-\infty}^\infty e^{-x^2}\,dx \quad$ [Solution]

Q9: $\displaystyle \int_0^\infty e^{-x^2}x^m\,dx=\frac{1}{2}\Gamma(\frac{m+1}{2})$ for $m>-1$.

Q10: Using the beta-gamma functions, find the following integrals:

1. $\displaystyle \int_0^{\frac{\pi}{2}} \sin^2 \theta \cos^3\theta\,d\theta$
2. $\displaystyle \int_0^{\frac{\pi}{2}} \cos^5 \theta \,d\theta$. [Note that $\displaystyle \int_0^{\frac{\pi}{2}} \sin^5 \theta \,d\theta$ = $\displaystyle \int_0^{\frac{\pi}{2}} \cos^5 \theta \,d\theta$. This is because B(m,n)=B(n,m)]

Q11: Show that the following:

1. Prove that $3 \cdot 6 \cdot 9 \cdots 3n=3^n \Gamma(n+1)$.
2. Prove that $B(m+1,n)+B(m,n+1)=B(m,n)$.

Q12: Define an orthogonal sequence. Prove that the sequence $\{\sin n\theta\}_{n=1}^\infty$ is an orthogonal sequence over the interval $(0, \pi)$ or $(-\pi, \pi)$.

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Laplace Transform

Q13: Find the Laplace transform:

1. $L\{\sin^2 t\}$
2. $L\{\cos^2 t -t^3 e^t\}$
3. $L \left\{\dfrac{e^{-5t}-e^{3t}}{t} \right\}$

Q14: State the first and second shifting property of Laplace transforms.

Q15: Find the inverse Laplace transforms:

1. $L^{-1} \left \{\dfrac{4s+3}{(s-1)^2(s+2)} \right \}$
2. $L^{-1}\left\{\dfrac{3s}{s^2-9}-\dfrac{1}{s-6}+\dfrac{1}{s^5} \right\}$

Q16: Using the concept of Laplace transform, evaluate $\displaystyle \int_0^\infty e^{-7t} \text{cosh}\, 3t \,dt$.

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Study Material:

Charlier Polynomials

Hermite Polynomials

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1. 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 $(1-x^2)^{-1/2}$.
2. Find the generating function for the Hermite polynomials 𝐻_n(𝑥). Compute 𝐻_2n(0) and 𝐻_2n+1(0).
3. Find the generating function for Charlier polynomials. Also, find its orthogonality relation.