---
title: "Define Beta and Gamma functions"
subject: "Mathematics I"
chapter: 2
key: "semester_1/mathematics_i:define-beta-gamma-functions"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/define-beta-gamma-functions
site: BCT Engineering
---

# Define Beta and Gamma functions

Mathematics I · Chapter 2: Integration and its Applications

- Asked in: 2074 Asoj; 2074 Chaitra, 5 marks; 2073 Shrawan; 2072 Chaitra (asked 4 times)
- Video: [Gamma Function - Properties & Its Application | Integral Calculus](https://www.youtube.com/watch?v=SduCO-r_TOA) — Dr.Gajendra Purohit and 2 more
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/define-beta-gamma-functions
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-2

## Question

Define Beta function and Gamma function.

## Short answer

Gamma function: Γ(n) = ∫₀^∞ e^{−x} x^{n−1} dx (n>0). Beta function: B(m,n) = ∫₀¹ x^{m−1}(1−x)^{n−1} dx (m,n>0); related by B(m,n) = Γ(m)Γ(n)/Γ(m+n).

## Answer

#### Gamma function
For n > 0, the Gamma function is defined by the improper integral

Γ(n) = ∫₀^∞ e^(−x) x^(n−1) dx.

Key properties:
- Γ(n + 1) = n Γ(n), so Γ(n + 1) = n! for a positive integer n.
- Γ(1) = 1, Γ(1/2) = √π.
- Γ(n)Γ(1 − n) = π / sin(nπ), 0 < n < 1.

#### Beta function
For m, n > 0, the Beta function (Eulerian integral of the first kind) is

B(m, n) = ∫₀¹ x^(m−1) (1 − x)^(n−1) dx.

Key properties:
- Symmetry: B(m, n) = B(n, m).
- Trigonometric form: B(m, n) = 2∫₀^(π/2) sin^(2m−1)θ cos^(2n−1)θ dθ.
- Relation with Gamma: B(m, n) = Γ(m) Γ(n) / Γ(m + n).

**Example:** B(2, 3) = Γ(2)Γ(3)/Γ(5) = (1·2)/24 = 1/12.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The Gamma function is defined for n>0 as
   A. integral from 0 to 1 of x^(n-1)(1-x)^(n-1) dx
   B. integral from 0 to ∞ of e^(-x) x^n dx
   C. integral from 0 to ∞ of e^(x) x^(n-1) dx
   D. integral from 0 to ∞ of e^(-x) x^(n-1) dx
   Answer: D. This improper integral is the Eulerian integral of the second kind. (Key point: Gamma definition)

2. Which holds for the Gamma function?
   A. Γ(n+1) = nΓ(n), so Γ(n+1) = n! for positive integers
   B. Γ(n+1) = Γ(n)+1
   C. Γ(n) = n!
   D. Γ(n+1) = (n+1)Γ(n)
   Answer: A. The recurrence gives factorials. (Key point: Gamma recurrence)

3. The Beta function B(m,n) is
   A. integral from 0 to ∞ of e^(-x) x^(m+n-1) dx
   B. integral from 0 to 1 of x^(m-1)(1-x)^(n-1) dx
   C. integral from 0 to 1 of x^m (1-x)^n dx
   D. integral from 0 to ∞ of x^(m-1)/(1+x)^n dx
   Answer: B. This is the Eulerian integral of the first kind. (Key point: Beta definition)

4. Beta and Gamma are related by
   A. B(m,n) = Γ(m)+Γ(n)-Γ(m+n)
   B. B(m,n) = Γ(m+n)/(Γ(m)Γ(n))
   C. B(m,n) = Γ(m)Γ(n)/Γ(m+n)
   D. B(m,n) = Γ(m)Γ(n)
   Answer: C. For example B(2,3) = Γ(2)Γ(3)/Γ(5) = 1/12. (Key point: Relation)

## Related in Mathematics I

- [Beta-Gamma: ∫ x⁵√(2ax−x²) dx](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Abeta-gamma-x5-sqrt-2ax-x2) — Mathematics I, chapter 2, asked 5×
- [Beta-Gamma: ∫ sin⁴x cos²x dx = (3π−4)/192](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Abeta-gamma-sin4-cos2-pi4) — Mathematics I, chapter 2, asked 2×
- [Beta-Gamma: ∫ cos²6θ sin⁴3θ dθ = 7π/192](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Abeta-gamma-cos2-6theta-sin4-3theta) — Mathematics I, chapter 2
- [Beta-Gamma: ∫ cos⁴3θ sin²6θ dθ = 5π/192](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Abeta-gamma-cos4-3theta-sin2-6theta) — Mathematics I, chapter 2, asked 3×
- [Gamma function: ∫ dx/(1−x⁶)^(1/6) = π/3](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Agamma-dx-over-1-minus-x6) — Mathematics I, chapter 2, asked 4×
- [Gamma function: ∫ sin⁶(x/2) cos⁶(x/2) dx](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Agamma-sin6-cos6-half-x) — Mathematics I, chapter 2, asked 3×