Morera's Theorem

Morera's Theorem In Complex Analysis :-

Morera's Theorem

Statement :

       If f(z) is continuous in a simple connected domain D and if 

                   ∫f(z) dz = 0

                   c

   for every closed path in D, then f(z) is analytic in D . 

Morera's Theorem

Proof :

Morera's Theorem

     Let z₀ be a fixed point and z a variable point inside the domain D , then the value of the integral

                   z

                   ∫ f(z) dz 

                  z₀

 is independent of the curve joining z₀ to z and is a function of the upper limit z . Then we have
                   z
        f(z) = ∫ f(t) dt .............(1)
                  z₀
                        z+h
then F(z+h) = ∫ f(t) dt
                        z₀
                                   z+h          z
Now F(z+h) - F(z) = ∫ f(t) dt - ∫ f(t) dt
                                  z₀             z₀
                                  z₀            z+h
                              = ∫ f(t) dt + ∫ f(t) dt
                                z+h            z₀
                                z+h
                             = ∫ f(t) dt             ........(2)
                                z

   Now the integral in (2) is independent of the path and may be taken along a straight line segment joining z to z+h .

Morera's Theorem

Hence [F(z+h) - F(z)]/ h  - f(z)
                    z+h
          = 1/h  ∫ f(t) dt - f(z)/h . h
                    z
                    z+h                 z+h
         = 1/h [ ∫ f(t) dt - f(z) ∫ dt ]
                     z                      z
                   z+h
         = 1/h ∫ [f(t) - f(z)] dt    ..........(3)
                  z

       Since f(t) is continuous at z , so given ε>0 ∃ δ>0 such that |f(t) - f(z)|<ε for every t satisfying |t-z|<δ .

    Let us choose h such that |h|<δ so that the relation |f(t) - f(z)| <ε is satisfied for every point t on the line segment joining z to z+h .

Morera's Theorem

     From equation (3) we get

|F'(z+h) - F(h) / h - f(z) |
                                             z+h
                               ≤  1/|h| ∫ |f(t) - f(z)||dt|
                                              z
                                                 z+h
                               < 1/|h| . ε ∫ |dt|
                                                  z
                               = 1/|h| ε |h|

                               = ε

         Since ε is arbitrary we get

            lim  F(z+h) - F(z) / h = f(z)
         h--->0

        ⇒F'(z) exists for all values of z in D and     F'(z) = f(z) .

       Hence F(z) is analytic in D .

   We know that derivative of an analytic function is analytic

       ⇒ f(z) is also analytic in D . (Proved)

About The Scientist :-

         Giacinto Morera (18 July 1856 – 8 February 1909), was an Italian engineer and mathematician. He is known for Morera's theorem in the theory of functions of a complex variable and for his work in the theory of linear elasticity. For More

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