{"id":5101,"date":"2021-08-31T11:24:38","date_gmt":"2021-08-31T05:54:38","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=5101"},"modified":"2026-02-06T05:08:52","modified_gmt":"2026-02-05T23:38:52","slug":"what-are-beta-and-gamma-functions","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-are-beta-and-gamma-functions\/","title":{"rendered":"What are beta and gamma functions?"},"content":{"rendered":"\n

Overview<\/strong><\/h2>\n\n\n\n

As we know, functions play an important role in mathematics. A function defines a specific correlation between the two data sets. Beta<\/a> and gamma<\/a> functions are from the group of Euler\u2019s integral functions.<\/p>\n\n\n\n

In this blog we will discuss the beta and gamma function, their properties, relationship between the two and problems based on these functions.<\/p>\n\n\n\n

The study of beta and gamma functions is very useful in the areas of mathematics and science.<\/p>\n\n\n\n

We use this concept in string theory, complex physics, integral calculus,  etc.<\/p>\n\n\n\n

Beta functions<\/strong><\/h2>\n\n\n\n

This is the first kind of Euler\u2019s integral, this function has real number domain,.<\/p>\n\n\n\n

The beta function<\/a> is-<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Gamma function<\/strong><\/h2>\n\n\n\n

The gamma function<\/a> is an improper integral, which is dependent on n,<\/p>\n\n\n\n

This is defined as the second kind of Euler\u2019s integral<\/p>\n\n\n

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Note- Beta function<\/a> is symmetrical with respect to m and n.<\/strong><\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Some important results-<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

The result <\/strong><\/strong> is known as the recurrence formula for gamma function.<\/strong><\/p>\n\n\n\n

Note-<\/strong><\/p>\n\n\n\n

1.<\/strong>     <\/strong>Symmetry- <\/strong><\/strong><\/p>\n\n\n\n

2.<\/strong>     <\/strong>Beta function in terms of trigonometric functions can be written as-<\/strong><\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Evaluation of beta function ? (m, n)<\/strong><\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Here we have-<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Or<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Again integrate by parts, we get-<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Repeating the process above, integrating by parts we get-<\/p>\n\n\n

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or<\/p>\n\n\n

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Evaluation of gamma function<\/a>–<\/strong><\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n
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Integrating by parts, we take as first function-<\/p>\n\n\n\n

We get<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Replace n by n+1,<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Example: Evaluate<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Sol:<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Example: Find the value of <\/strong><\/strong><\/p>\n\n\n\n

Sol:<\/strong><\/p>\n\n\n\n

We know that<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

So that<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Interested in learning about similar topics? https:\/\/thechrysaliscapital.com\/team\/<\/a> Here are a few hand-picked blogs for you!<\/p>\n\n\n\n