Mathquery

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