Special Functions Practice Problems

Special functions practice problems. A list of practice problems on Special Functions are given here.

Problems on Beta Gamma Functions

Q1: For what values of m and n, the integral $\displaystyle \int_0^1 x^{m+1}(1-x)^{n-3}\,dx \quad$ converges.

Q2: Find the Value of $B\left(\dfrac{1}{2}, \dfrac{1}{2} \right)\quad$ [Solution]

Q3: What are the values of $\Gamma(1)$ and $\Gamma(5)$?

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

Q5: Compute $\Gamma\left(\dfrac{3}{2}\right) \Gamma\left(\dfrac{5}{2}\right)$

Q6: Evaluate: $\displaystyle \int_0^\infty e^{-x^2}\,dx \quad$ [Solution] and $\displaystyle \int_{-\infty}^\infty e^{-x^2}\,dx \quad$ [Solution]

Q7: Evaluate $\displaystyle \int_0^{\frac{\pi}{2}} \sqrt{\tan x}\,dx$. Is it same as $\displaystyle \int_0^{\frac{\pi}{2}} \sqrt{\cot x}\,dx$? Give reasons.

Q8: Using the beta-gamma functions, find $\displaystyle \int_0^{\frac{\pi}{2}} \sin^3x \cos^4x\,dx$.

Problems on Orthogonality Conditions

Q1: Define an orthogonal sequence. Show that the sequence $\{\cos n\theta\}_{n=1}^\infty$ is an orthogonal sequence over the interval $(0, \pi)$.

Q2: Define an orthogonal polynomial sequence with respect to the weight function W(x). Prove 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 $(1-x^2)^{-1/2}$.