Fourier sine and cosine series explained

In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after Joseph Fourier.

Notation

In this article, denotes a real-valued function on

R which is periodic with period 2L.

Sine series

If is an odd function with period

2 L , then the Fourier Half Range sine series of f is defined to be$$f(x)\; =\; \backslash sum\_^\backslash infty\; b\_n\; \backslash sin\; \backslash frac$$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$$b\_n\; =\; \backslash frac\; \backslash int\_0^L\; f(x)\; \backslash sin\; \backslash frac\; \backslash ,\; dx,\; \backslash quad\; n\; \backslash in\; \backslash mathbb\; .$$

Cosine series

If is an even function with a period

2 L , then the Fourier cosine series is defined to be$$f(x)\; =\; \backslash frac\; +\; \backslash sum\_^\; c\_n\; \backslash cos\; \backslash frac$$where$$c\_n\; =\; \backslash frac\; \backslash int\_0^L\; f(x)\; \backslash cos\; \backslash frac\; \backslash ,\; dx,\; \backslash quad\; n\; \backslash in\; \backslash mathbb\_0\; .$$

Remarks

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

See also

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

Bibliography

• Book: Byerly , William Elwood . An Elementary Treatise on Fourier's Series: And Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. 2. Ginn. 1893. Chapter 2: Development in Trigonometric Series . https://books.google.com/books?id=BMQ0AQAAMAAJ&pg=PA30. 30.
• Book: Carslaw , Horatio Scott . Introduction to the Theory of Fourier's Series and Integrals, Volume 1. 2. Macmillan and Company. 1921. Chapter 7: Fourier's Series . https://books.google.com/books?id=JNVAAAAAIAAJ&pg=PA196. 196.