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Mittag-Leffler's Theorem


If a function analytic at the origin has no singularities other than poles for finite x, and if we can choose a sequence of contours C_m about z=0 tending to infinity such that |f(z)| never exceeds a given quantity M on any of these contours and int|dz/z| is uniformly bounded on them, then

 f(z)=f(0)+lim[P_m(z)-P_m(0)],

where P_m(z) is the sum of the principal parts of f(z) at all poles alpha within C_m. If there is a pole at z=0, then we can replace f(0) by the negative powers and the constant term in the Laurent series of f(z) about z=0.


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References

Jeffreys, H. and Jeffreys, B. S. "Mittag-Leffler's Theorem." §12.006 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 383-386, 1988.

Referenced on Wolfram|Alpha

Mittag-Leffler's Theorem

Cite this as:

Weisstein, Eric W. "Mittag-Leffler's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Mittag-LefflersTheorem.html

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