Improper Integrals

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 7.8 – Improper Integrals

Gamma Function

Definition: Gamma Function

${\displaystyle \mathrm{\Gamma }(n+1)={\int }_0^{\mathrm{\infty }}{x}^n{e}^{-x}\text{d}x}$$\displaystyle \Gamma(n+1)=\int_0^{\infty}{x^ne^{-x}\,\text{d}x}$

Investigation 09

Compute the following:

1. ${\displaystyle \mathrm{\Gamma }(1)={\int }_0^{\mathrm{\infty }}{x}^0{e}^{-x}\text{d}x}$$\displaystyle \Gamma(1)=\int_0^{\infty}{x^0 e^{-x}\,\text{d}x}$
2. ${\displaystyle \mathrm{\Gamma }(2)={\int }_0^{\mathrm{\infty }}{x}^1{e}^{-x}\text{d}x}$$\displaystyle \Gamma(2)=\int_0^{\infty}{x^1 e^{-x}\,\text{d}x}$
3. ${\displaystyle \mathrm{\Gamma }(3)={\int }_0^{\mathrm{\infty }}{x}^2{e}^{-x}\text{d}x}$$\displaystyle \Gamma(3)=\int_0^{\infty}{x^2 e^{-x}\,\text{d}x}$
4. ${\displaystyle \mathrm{\Gamma }(5)={\int }_0^{\mathrm{\infty }}{x}^4{e}^{-x}\text{d}x}$$\displaystyle \Gamma(4)=\int_0^{\infty}{x^3 e^{-x}\,\text{d}x}$
5. ${\displaystyle \mathrm{\Gamma }(5)={\int }_0^{\mathrm{\infty }}{x}^4{e}^{-x}\text{d}x}$$\displaystyle \Gamma(5)=\int_0^{\infty}{x^4 e^{-x}\,\text{d}x}$
6. ${\displaystyle \mathrm{\Gamma }(6)={\int }_0^{\mathrm{\infty }}{x}^5{e}^{-x}\text{d}x}$$\displaystyle \Gamma(6)=\int_0^{\infty}{x^5 e^{-x}\,\text{d}x}$
7. ${\displaystyle \mathrm{\Gamma }(7)={\int }_0^{\mathrm{\infty }}{x}^6{e}^{-x}\text{d}x}$$\displaystyle \Gamma(7)=\int_0^{\infty}{x^6 e^{-x}\,\text{d}x}$

xxxxxxxxxx
1
1
(* To compute the Gamma function *)
2
(* Example: Gamma(4) *)
3

4
Gamma[4]

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Sunday, 12 June 2022 - 20:30 (EDT)