Editing Morera's theorem (section)

==Applications==
Morera's theorem is a standard tool in [[complex analysis]].  It is used in almost any argument that involves a non-algebraic construction of a holomorphic function.

===Uniform limits===
For example, suppose that ''f''<sub>1</sub>,&nbsp;''f''<sub>2</sub>,&nbsp;... is a sequence of holomorphic functions, [[uniform convergence|converging uniformly]] to a continuous function ''f'' on an open disc.  By [[Cauchy's integral theorem|Cauchy's theorem]], we know that
<math display="block">\oint_C f_n(z)\,dz = 0</math>
for every ''n'', along any closed curve ''C'' in the disc.  Then the uniform convergence implies that
<math display="block">\oint_C f(z)\,dz = \oint_C \lim_{n\to \infty} f_n(z)\,dz =\lim_{n\to \infty} \oint_C f_n(z)\,dz = 0 </math>
for every closed curve ''C'', and therefore by Morera's theorem ''f'' must be holomorphic.  This fact can be used to show that, for any [[open set]] {{math|Ω ⊆ '''C'''}}, the set {{math|''A''(Ω)}} of all [[bounded function|bounded]], analytic functions {{math|''u'' : Ω → '''C'''}} is a [[Banach space]] with respect to the [[supremum norm]].

===Infinite sums and integrals===
Morera's theorem can also be used in conjunction with [[Fubini's theorem]] and the [[Weierstrass M-test]] to show the analyticity of functions defined by sums or integrals, such as the [[Riemann zeta function]]
<math display="block">\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}</math>
or the [[Gamma function]]
<math display="block">\Gamma(\alpha) = \int_0^\infty x^{\alpha-1} e^{-x}\,dx.</math>

Specifically one shows that
<math display="block"> \oint_C \Gamma(\alpha)\,d\alpha = 0 </math>
for a suitable closed curve ''C'', by writing
<math display="block"> \oint_C \Gamma(\alpha)\,d\alpha = \oint_C \int_0^\infty x^{\alpha-1} e^{-x} \, dx \,d\alpha </math>
and then using Fubini's theorem to justify changing the order of integration, getting
<math display="block"> \int_0^\infty \oint_C x^{\alpha-1} e^{-x} \,d\alpha \,dx = \int_0^\infty e^{-x} \oint_C x^{\alpha-1} \, d\alpha \,dx. </math>

Then one uses the analyticity of {{math|''α'' ↦ ''x''<sup>''α''−1</sup>}} to conclude that
<math display="block"> \oint_C x^{\alpha-1} \, d\alpha = 0, </math>
and hence the double integral above is&nbsp;0.  Similarly, in the case of the zeta function, the M-test justifies interchanging the integral along the closed curve and the sum.