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As the negative degree of the Laurent series rises, it approaches the correct function. Laurent series with complex coefficients are an important tool in complex analysis, especially to investigate the behavior of functions near singularities.
The residue Res(f, c) of f at c is the coefficient a −1 of (z − c) −1 in the Laurent series expansion of f around c. Various methods exist for calculating this value, and the choice of which method to use depends on the function in question, and on the nature of the singularity.
Complex analysis, traditionally known as the theory of functions of a complex variable, ... Laurent series are the complex-valued equivalent to Taylor series, ...
Mathematical analysis → Complex analysis: Complex analysis; Complex numbers ... a nonzero meromorphic function f is the sum of a Laurent series with at most finite ...
For type I, the singularities are (movable) double poles of residue 0, and the solutions all have an infinite number of such poles in the complex plane. The functions with a double pole at z 0 {\displaystyle z_{0}} have the Laurent series expansion
The principal part at = of a function = = ()is the portion of the Laurent series consisting of terms with negative degree. [1] That is, = is the principal part of at .If the Laurent series has an inner radius of convergence of , then () has an essential singularity at if and only if the principal part is an infinite sum.
In complex analysis, a partial fraction expansion is a way of writing a meromorphic function as an infinite sum of rational functions and polynomials. When f ( z ) {\displaystyle f(z)} is a rational function, this reduces to the usual method of partial fractions .
As in complex analysis of functions of one variable, which is the case n = 1, the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables z i. Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the n-dimensional Cauchy–Riemann equations.