{"id":4483,"date":"2021-08-03T10:13:07","date_gmt":"2021-08-03T04:43:07","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=4483"},"modified":"2025-12-22T09:46:01","modified_gmt":"2025-12-22T04:16:01","slug":"what-is-the-periodic-function","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-the-periodic-function\/","title":{"rendered":"What is the periodic function?"},"content":{"rendered":"\n

Overview<\/strong>(periodic function)<\/h2>\n\n\n\n

A function f(x) is said to be periodic function<\/a> if f(x + T) = f(x) for all real x and some positive number T, here T is the period of f(x).<\/p>\n\n\n\n

Suppose if we take sin x, then it repeats its value after the period of 2pi<\/p>\n\n\n\n

such that, we write this as<\/p>\n\n\n

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We can say that sin x is a periodic function with the period of .<\/p>\n\n\n\n

sin x, cos x, sec x, cosec x are the periodic functions with period , where tan x and cot x are the periodic functions with period<\/p>\n\n\n\n

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

1.     Sin x is also known as sinusoidal periodic function.<\/p>\n\n\n\n

2.     The period of a sum of a number of periodic functions is the least common multiple of the periods.<\/p>\n\n\n\n

3.     A constant function is periodic for any positive T.<\/p>\n\n\n\n

4.     Suppose T is the period of f(x) then nT is also periodic of f for any integer n.<\/p>\n\n\n\n

Fourier series<\/strong><\/h2>\n\n\n\n

Trigonometric series-<\/p>\n\n\n\n

This is a functional series of the form,<\/p>\n\n\n

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

The constants a0, an and bn are the coefficients.<\/p>\n\n\n\n

Fourier series of a periodic function f(x) with period is the trigonometric series with Fourier coefficients  a0, an and bn<\/p>\n\n\n\n

Any periodic function can be expanded in the form of Fourier series<\/a>.<\/p>\n\n\n\n

How to determine <\/strong><\/p>\n\n\n\n

We know that, the Fourier series,<\/strong><\/p>\n\n\n

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

To find <\/strong><\/p>\n\n\n\n

Intergrate equation (1) on both sides, from 0 to 2\u03c0<\/strong><\/p>\n\n\n

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That gives<\/strong><\/p>\n\n\n

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To find <\/strong><\/p>\n\n\n\n

<\/strong><\/p>\n\n\n\n

Multiply each side of eq. (1) by cos nx and integrate from 0 to 2<\/strong>\u03c0<\/strong><\/p>\n\n\n\n

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

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Similarly we can find <\/strong><\/strong> by, multiplying eq. (1) by sin nx and integrating from 0 to  2<\/strong>\u03c0<\/strong><\/p>\n\n\n

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Solved example<\/strong><\/h2>\n\n\n\n

Example: Find the fourier series of the function f(x) = x  where 0 < x < 2 <\/strong>\u03c0<\/strong><\/p>\n\n\n\n

Sol. <\/strong>We know that, from Fourier series,<\/p>\n\n\n

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First we will find a_0<\/p>\n\n\n

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Now a_n<\/p>\n\n\n

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And<\/p>\n\n\n

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Now put the value in Fourier series, we get<\/p>\n\n\n

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Example: Find the Fourier series for f(x) = <\/strong><\/strong> in the interval <\/strong><\/strong><\/p>\n\n\n\n

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

Suppose <\/p>\n\n\n

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Then<\/p>\n\n\n\n

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

And<\/p>\n\n\n\n

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

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

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

And then<\/p>\n\n\n\n

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

Now put these value in equations (1), we get-<\/p>\n\n\n\n

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

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