Fourier sine and cosine series

In this article, {{math|f}} denotes a real-valued function on $\backslash mathbb\{R\}$ which is periodic with period 2L.

If {{math|f}} is an odd function with period $2L$, then the Fourier Half Range sine series of f is defined to be

$$f(x)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; b\_n\; \backslash sin\; \backslash left(\backslash frac\{n\backslash pi\; x\}\{L\}\backslash right)$$

which is just a form of complete Fourier series with the only difference that $a\_0$ and $a\_n$ are zero, and the series is defined for half of the interval.

In the formula we have

$$b\_n\; =\; \backslash frac\{2\}\{L\}\; \backslash int\_0^L\; f(x)\; \backslash sin\; \backslash left(\backslash frac\{n\backslash pi\; x\}\{L\}\backslash right)\; \backslash ,\; dx,\; \backslash quad\; n\; \backslash in\; \backslash mathbb\{N\}\; .$$

If {{math|f}} is an even function with a period $2L$, then the Fourier cosine series is defined to be

$$f(x)\; =\; \backslash frac\{a\_0\}\{2\}\; +\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; a\_n\; \backslash cos\; \backslash left(\backslash frac\{n\; \backslash pi\; x\}\{L\}\backslash right)$$

where

$$a\_n\; =\; \backslash frac\{2\}\{L\}\; \backslash int\_0^L\; f(x)\; \backslash cos\; \backslash left(\backslash frac\{n\backslash pi\; x\}\{L\}\backslash right)\; \backslash ,\; dx,\; \backslash quad\; n\; \backslash in\; \backslash mathbb\{N\}\_0\; .$$

This notion can be generalized to functions which are not even or odd, but then the above formulas will look different.

• Fourier series
• Fourier analysis
• Least-squares spectral analysis

• {{cite book

|first=William Elwood |last=Byerly

|title=An Elementary Treatise on Fourier's Series: And Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics

|edition=2

|publisher=Ginn

|date=1893

|chapter=Chapter 2: Development in Trigonometric Series |chapter-url=https://books.google.com/books?id=BMQ0AQAAMAAJ&pg=PA30

|page=30

}}

• {{cite book

|first=Horatio Scott |last=Carslaw

|title=Introduction to the Theory of Fourier's Series and Integrals, Volume 1

|edition=2

|publisher=Macmillan and Company

|date=1921

|chapter=Chapter 7: Fourier's Series |chapter-url=https://books.google.com/books?id=JNVAAAAAIAAJ&pg=PA196

|page=196

}}

Category:Fourier series