Morera’s Theorem

Morera’s Theorem is regarded as the converse of Cauchy-Goursat Theorem, which states that if a function f(z) is analytic and single-valued inside and on a simple closed contour C, then

Morera’s theorem was named after the Italian engineer and mathematician Giacinto Morera, known for his work in Linear Elasticity and the theory of functions in complex variables.

Statement of Morera’s Theorem

According to Morera’s theorem for integration of complex functions in a simply connected domain D, as follows –

A continuous function f(z) defined in a simply connected domain D, whose closed curve integration over every closed contour C in the domain D must vanish, for f to be analytic or holomorphic within D.

Proof of Morera’s Theorem

As per the statement of the theorem we have a continuous function f defined in a simply connected domain D and

Let z + h be any point in the neighbourhood of z, then we have

Subtracting (2) from (1), we get

Since the integral in (3) is path-independent and without loss of generality, it may be taken along line segment joining points z and z + h. Now,

Since f(𝜁) is continuous at z in D, by the definition of continuous function, for any given ε > 0, there exist a positive number δ such that

|f(𝜁) – f(z)| < ε for all 𝜁 satisfying |𝜁 – z| < δ ………..(5)

Now if 𝜁 = z + h then | z + h – z| < δ ⇒ |h| < δ. Then the inequality (5) is satisfied for every 𝜁 on the line segment joining the points z and z + h. Hence from (4) and (5) we have

Since ε is arbitrary, we have from (6)

Which means F(z) is differentiable, hence analytic with the derivative f(z). That is, F’(z) = f(z) for every z in D. Consequently F(z) is analytic within D. But the derivative of an analytic function is also analytic. It follows that f(z) is also analytic in D.

Important Facts on Morera’s Theorem

• The given function f is continuous in the domain D.
• f(z) exists for every z in D.
• If C is any closed curve within domain D for which the given function converges within C then
  \(\begin{array}{l}\int _{C}f(z) dz = 0\end{array} \)
  .
• Consequently, f is analytic within D.

Related Articles

• Complex Numbers
• Analytic Functions
• Limits and Continuity
• Differentiability
• Integration
• Complex Conjugate

Solved Example on Morera’s Theorem

Example :

Show that the function f defined by

Solution:

This could be easily proved by applying Morera’s Theorem. We have a continuous function of two variables, by changing the order of integration we can easily apply Morera’s Theorem.

Now,

That is, the integral is absolute convergent and |f(z)| < 1/|x|

Consider

where C is a closed contour in D.

Since, the given integral converges, we can have

As e^zt/(t + 1) is analytic inside C.

By Morera’s Theorem f is analytic within D.