Mittag-Leffler's theorem

Mittag-Leffler's theorem In complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed poles. Conversely, it can be used to express any meromorphic function as a sum of partial fractions. It is sister to the Weierstrass factorization theorem, which asserts existence of holomorphic functions with prescribed zeros.

The theorem is named after the Swedish mathematician Gösta Mittag-Leffler who published versions of the theorem in 1876 and 1884.[1][2][3] Contents 1 Theorem 1.1 Proof sketch 2 Example 3 Pole expansions of meromorphic functions 4 See also 5 References 6 External links Theorem Let {displaystyle U} be an open set in {displaystyle mathbb {C} } and {displaystyle Esubset U} be a subset whose limit points, if any, occur on the boundary of {displaystyle U} . For each {displaystyle a} in {displaystyle E} , let {displaystyle p_{a}(z)} be a polynomial in {displaystyle 1/(z-a)} without constant coefficient, i.e. of the form {displaystyle p_{a}(z)=sum _{n=1}^{N_{a}}{frac {c_{a,n}}{(z-a)^{n}}}.} Then there exists a meromorphic function {displaystyle f} on {displaystyle U} whose poles are precisely the elements of {displaystyle E} and such that for each such pole {displaystyle ain E} , the function {displaystyle f(z)-p_{a}(z)} has only a removable singularity at {displaystyle a} ; in particular, the principal part of {displaystyle f} at {displaystyle a} is {displaystyle p_{a}(z)} . Furthermore, any other meromorphic function {displaystyle g} on {displaystyle U} with these properties can be obtained as {displaystyle g=f+h} , where {displaystyle h} is an arbitrary holomorphic function on {displaystyle U} .

Proof sketch One possible proof outline is as follows. If {displaystyle E} is finite, it suffices to take {textstyle f(z)=sum _{ain E}p_{a}(z)} . If {displaystyle E} is not finite, consider the finite sum {textstyle S_{F}(z)=sum _{ain F}p_{a}(z)} where {displaystyle F} is a finite subset of {displaystyle E} . While the {displaystyle S_{F}(z)} may not converge as F approaches E, one may subtract well-chosen rational functions with poles outside of {displaystyle U} (provided by Runge's theorem) without changing the principal parts of the {displaystyle S_{F}(z)} and in such a way that convergence is guaranteed.

Example Suppose that we desire a meromorphic function with simple poles of residue 1 at all positive integers. With notation as above, letting {displaystyle p_{k}(z)={frac {1}{z-k}}} and {displaystyle E=mathbb {Z} ^{+}} , Mittag-Leffler's theorem asserts (non-constructively) the existence of a meromorphic function {displaystyle f} with principal part {displaystyle p_{k}(z)} at {displaystyle z=k} for each positive integer {displaystyle k} . More constructively we can let {displaystyle f(z)=zsum _{k=1}^{infty }{frac {1}{k(z-k)}}.} This series converges normally on {displaystyle mathbb {C} } (as can be shown using the M-test) to a meromorphic function with the desired properties.

Pole expansions of meromorphic functions Here are some examples of pole expansions of meromorphic functions: {displaystyle tan(z)=sum _{n=0}^{infty }{frac {8z}{(2n+1)^{2}pi ^{2}-4z^{2}}}} {displaystyle csc(z)=sum _{nin mathbb {Z} }{frac {(-1)^{n}}{z-npi }}={frac {1}{z}}+2zsum _{n=1}^{infty }(-1)^{n}{frac {1}{z^{2}-(n,pi )^{2}}}} {displaystyle sec(z)equiv -csc left(z-{frac {pi }{2}}right)=sum _{nin mathbb {Z} }{frac {(-1)^{n-1}}{z-left(n+{frac {1}{2}}right)pi }}=sum _{n=0}^{infty }{frac {(-1)^{n}(2n+1)pi }{(n+{frac {1}{2}})^{2}pi ^{2}-z^{2}}}} {displaystyle cot(z)equiv {frac {cos(z)}{sin(z)}}=lim _{Nto infty }sum _{n=-N}^{N}{frac {1}{z-npi }}={frac {1}{z}}+2zsum _{k=1}^{infty }{frac {1}{z^{2}-(k,pi )^{2}}}} {displaystyle csc ^{2}(z)=sum _{nin mathbb {Z} }{frac {1}{(z-n,pi )^{2}}}} {displaystyle sec ^{2}(z)={frac {d}{dz}}tan(z)=sum _{n=0}^{infty }{frac {8((2n+1)^{2}pi ^{2}+4z^{2})}{((2n+1)^{2}pi ^{2}-4z^{2})^{2}}}} {displaystyle {frac {1}{zsin(z)}}={frac {1}{z^{2}}}+sum _{nneq 0}{frac {(-1)^{n}}{pi n(z-pi n)}}={frac {1}{z^{2}}}+sum _{n=1}^{infty }{(-1)^{n}}{frac {2}{z^{2}-(n,pi )^{2}}}} See also Riemann–Roch theorem Liouville's theorem Mittag-Leffler condition of an inverse limit Mittag-Leffler summation Mittag-Leffler function References ^ Mittag-Leffler (1876). "En metod att analytiskt framställa en funktion af rational karakter, hvilken blir oändlig alltid och endast uti vissa föreskrifna oändlighetspunkter, hvilkas konstanter äro påförhand angifna". Öfversigt af Kongliga Vetenskaps-Akademiens förhandlingar Stockholm. 33 (6): 3–16. ^ Mittag-Leffler (1884). "Sur la représentation analytique des fonctions monogènes uniformes dʼune variable indépendante". Acta Mathematica. 4: 1–79. ^ Turner, Laura E. (2013-02-01). "The Mittag-Leffler Theorem: The origin, evolution, and reception of a mathematical result, 1876–1884". Historia Mathematica. 40 (1): 36–83. doi:10.1016/j.hm.2012.10.002. ISSN 0315-0860. Ahlfors, Lars (1953), Complex analysis (3rd ed.), McGraw Hill (published 1979), ISBN 0-07-000657-1. Conway, John B. (1978), Functions of One Complex Variable I (2nd ed.), Springer-Verlag, ISBN 0-387-90328-3. External links "Mittag-Leffler theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Categories: Theorems in complex analysis