Cauchy’s Theorem

Cauchy’s Theorem:

Let R be the closed region consisting of all points interior to and on the simple closed contour C.

If f is analytic in R and f' is continuous in R,

\int_C f(z)\,dz=0

(This is the precursor of Cauchy-Goursat Theorem, which allows us to drop the condition that f' is continuous.)

Proof Using Green’s Theorem:

Let C denote a positively oriented simple closed contour z=z(t), a\leq t\leq b.

\int_C f(z)\,d(z)=\int_a^b f[z(t)]z'(t)\,dt where f(z)=u(x,y)+iv(x,y) and z(t)=x(t)+iy(t). Thus

\begin{aligned} \int_C f(z)\,dz&=\int_a^b (u+iv)(x'+iy')\,dt\\    &=\int_a^b (ux'-vy')\,dt+i\int_a^b (vx'+uy')\,dt\\    &=\int_C u\,dx-v\,dy+i\int_C v\,dx+u\,dy    \end{aligned}

Next, we need Green’s Theorem:

\int_C P\,dx+Q\,dy=\iint_R (Q_x-P_y)\,dA

By assumption f' is continuous in R, thus the first-order partial derivatives of u and v are also continous. This is exactly what we need for Green’s Theorem.

Continuing from above, we get \int_C f(z)\,dz=\iint_R (-v_x-u_y)\,dA+i\iint_R(u_x-v_y)\,dA which is exactly zero in view of the Cauchy-Riemann equations u_x=v_y, u_y=-v_x!

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