Hurwitz's Theorem (complex Analysis)

In mathematics and in particular the field of complex analysis, Hurwitz's theorem is a theorem associating the zeroes of a sequence of holomorphic, compact locally uniformly convergent functions with that of their corresponding limit. The theorem is named after Adolf Hurwitz.

Statement

Let be a sequence of holomorphic functions on a connected open set ''G'' that converge uniformly on compact subsets of ''G'' to a holomorphic function ''f'' which is not constantly zero on ''G''. If ''f'' has a zero of order ''m'' at ''z''₀ then for every small enough ''ρ'' > 0 and for sufficiently large ''k'' ∈ N (depending on ''ρ''), ''f_k'' has precisely ''m'' zeroes in the disk defined by , ''z'' − ''z''₀,  < ''ρ'', including multiplicity. Furthermore, these zeroes converge to ''z''₀ as ''k'' → ∞.,

Remarks

The theorem does not guarantee that the result will hold for arbitrary disks. Indeed, if one chooses a disk such that ''f'' has zeroes on its boundary, the theorem fails. An explicit example is to consider the unit disk D and the sequence defined by

:$f_n(z) = z-1+\frac 1 n, \qquad z \in \mathbb C$

which converges uniformly to ''f''(''z'') = ''z'' − 1. The function ''f''(''z'') contains no zeroes in D; however, each ''f_n'' has exactly one zero in the disk corresponding to the real value 1 − (1/''n'').

Applications

Hurwitz's theorem is used in the proof of the Riemann mapping theorem, and also has the following two corollaries as an immediate consequence:
* Let ''G'' be a connected, open set and a sequence of holomorphic functions which converge uniformly on compact subsets of ''G'' to a holomorphic function ''f''. If each ''f_n'' is nonzero everywhere in ''G'', then ''f'' is either identically zero or also is nowhere zero.
* If is a sequence of univalent functions on a connected open set ''G'' that converge uniformly on compact subsets of ''G'' to a holomorphic function ''f'', then either ''f'' is univalent or constant.

Proof

Let ''f'' be an analytic function on an open subset of the complex plane with a zero of order ''m'' at ''z''₀, and suppose that is a sequence of functions converging uniformly on compact subsets to ''f''. Fix some ''ρ'' > 0 such that ''f''(''z'') ≠ 0 in 0 < , ''z'' − ''z''₀, ≤ ρ. Choose δ such that , ''f''(''z''),  > ''δ'' for ''z'' on the circle , ''z'' − ''z''₀,  = ''ρ''. Since ''f_k''(''z'') converges uniformly on the disc we have chosen, we can find ''N'' such that , ''f_k''(''z''),  ≥ ''δ''/2 for every ''k'' ≥ ''N'' and every ''z'' on the circle, ensuring that the quotient ''f_k''′(''z'')/''f_k''(''z'') is well defined for all ''z'' on the circle , ''z'' − ''z''₀,  = ''ρ''. By Weierstrass's theorem we have $f_k' \to f'$ uniformly on the disc, and hence we have another uniform convergence:

: $\frac \to \frac.$

Denoting the number of zeros of ''f_k''(''z'') in the disk by ''N_k'', we may apply the argument principle to find

: $m = \frac 1 \int_ \frac \,dz = \lim_ \frac 1 \int_ \frac \, dz = \lim_ N_k$

In the above step, we were able to interchange the integral and the limit because of the uniform convergence of the integrand. We have shown that ''N_k'' → ''m'' as ''k'' → ∞. Since the ''N_k'' are integer valued, ''N_k'' must equal ''m'' for large enough ''k''.

See also

*Rouché's theorem

References

*
*
* John B. Conway. ''Functions of One Complex Variable I''. Springer-Verlag, New York, New York, 1978.
* E. C. Titchmarsh, ''The Theory of Functions'', second edition (Oxford University Press, 1939; reprinted 1985), p. 119.
*{{Springer , title=Hurwitz theorem , id=H/h048160 , first=E.D. , last=Solomentsev

Theorems in complex analysis