Mittag-Leffler Expansions of Meromorphic Functions

Consider a meromorphic function whose only singularities are simple poles at , , , … () with residues , , , … at these poles. Then , a result known as the Mittag-Leffler (M-L) expansion [1, 2]. This Demonstration considers M-L expansions for the gamma function and several trigonometric functions with poles in the complex plane. You can construct explicit approximations for up to eight pairs of positive and negative poles and then display the full formula as . A 3D plot of the function on the complex plane is also shown, with the poles appearing as spikes; the argument of the complex function is color-coded.

An early motivation for expansions using partial fractions was a result derived by Euler in Introductio in analysin infinitorum (1748):

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The M-L expansion has proven to be extremely useful in many applications to mathematical physics. The author's derivation for a closed form for the Coulomb propagator [3] was based on an M-L expansion of Green's function.

References

[1] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., London: Cambridge University Press, 1958 pp. 134ff.

[2] G. Mittag-Leffler, "En metod att analytisk framställa en funktion at rationel karacte...," Öfversigt Kongl. Vetenskaps-Akademiens Förhandlinger, 33, 1876 pp. 3–16.

[3] S. M. Blinder, "Analytic Form for the Nonrelativistic Coulomb Propagator," Physical Review A, 43(1), 1991 pp. 13–16. doi:10.1103/PhysRevA.43.13.