In mathematics, the Laurent series of a complex function f ( z ) {\displaystyle f(z)} is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass had previously described it in a paper written in 1841 but not published until 1894.
Definition The Laurent series for a complex function f ( z ) {\displaystyle f(z)} about an arbitrary point c {\displaystyle c} is given by
f ( z ) = ∑ n = − ∞ ∞ a n ( z − c ) n , {\displaystyle f(z)=\sum _{n=-\infty }^{\infty }a_{n}(z-c)^{n},}
where the coefficients a n {\displaystyle a_{n}} are defined by a contour integral that generalizes Cauchy's integral formula:
a n = 1 2 π i ∮ γ f ( z ) ( z − c ) n + 1 d z . {\displaystyle a_{n}={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{(z-c)^{n+1}}}\,dz.}
The path of integration γ {\displaystyle \gamma } is counterclockwise around a Jordan curve enclosing c {\displaystyle c} and lying in an annulus A {\displaystyle A} in which f ( z ) {\displaystyle f(z)} is holomorphic (analytic). The expansion for f ( z ) {\displaystyle f(z)} will then be valid anywhere inside the annulus. The annulus is shown in red in the figure on the right, along with an example of a suitable path of integration labeled γ {\displaystyle \gamma } . When γ {\displaystyle \gamma } is defined as the circle | z − c | = ϱ {\displaystyle |z-c|=\varrho } , where r < ϱ < R {\displaystyle r<\varrho <R} , this amounts to computing the complex Fourier coefficients of the restriction of f {\displaystyle f} to γ {\displaystyle \gamma } . The fact that these integrals are unchanged by a deformation of the contour γ {\displaystyle \gamma } is an immediate consequence of Cauchy's integral theorem. One may also obtain the Laurent series for a complex function f ( z ) {\displaystyle f(z)} at z = ∞ {\displaystyle z=\infty } . However, this is the same as when R → ∞ {\displaystyle R\rightarrow \infty } . The above integral formula may not offer the most practical method for computing the coefficients
a n {\displaystyle a_{n}} for a given function f ( z ) {\displaystyle f(z)} ; instead, one often pieces together the Laurent series by combining known Taylor expansions. Because the Laurent expansion of a function is unique whenever it exists, any expression of this form that equals the given function
f ( z ) {\displaystyle f(z)} in some annulus must actually be the Laurent expansion of f ( z ) {\displaystyle f(z)} .
Convergence
Laurent series with complex coefficients are an important tool in complex analysis, especially to investigate the behavior of functions near singularities. Consider for instance the function f ( x ) = e − 1 / x 2 {\displaystyle f(x)=e^{-1/x^{2}}} with f ( 0 ) = 0 {\displaystyle f(0)=0} . As a real function, it is infinitely differentiable everywhere; as a complex function however it is not differentiable at x = 0 {\displaystyle x=0} . The Laurent series of f ( x ) {\displaystyle f(x)} is obtained via the power series representation,
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