HOME

TheInfoList



Click Here for Items Related To - Conjugate Fourier Series
OR:
Conjugate Fourier Series on:   [Wikipedia]   [Google]   [Amazon]

In the mathematical field of
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...
, the conjugate Fourier series arises by realizing the Fourier series formally as the boundary values of the real part of a
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
on the unit disc. The imaginary part of that function then defines the conjugate series. studied the delicate questions of convergence of this series, and its relationship with the Hilbert transform. In detail, consider a
trigonometric series In mathematics, a trigonometric series is a infinite series of the form : \frac+\displaystyle\sum_^(A_ \cos + B_ \sin), an infinite version of a trigonometric polynomial. It is called the Fourier series of the integrable function f if the term ...
of the form :f(\theta) = \tfrac12 a_0 + \sum_^\infty \left(a_n\cos n\theta + b_n\sin n\theta\right) in which the coefficients ''a''''n'' and ''b''''n'' are
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s. This series is the real part of the
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
:F(z) = \tfrac12 a_0 + \sum_^\infty (a_n-ib_n)z^n along the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
with z=e^. The imaginary part of ''F''(''z'') is called the conjugate series of ''f'', and is denoted :\tilde(\theta) = \sum_^\infty \left(a_n\sin n\theta - b_n\cos n\theta\right).


See also

*
Harmonic conjugate In mathematics, a real-valued function u(x,y) defined on a connected open set \Omega \subset \R^2 is said to have a conjugate (function) v(x,y) if and only if they are respectively the real and imaginary parts of a holomorphic function f(z) of th ...


References

* * Fourier analysis Fourier series {{mathanalysis-stub