Cauchy's theorem

"Figure 1: a simple closed contour C in the complex z plane"

Here we follow standard texts, such as Spiegel (1964) or Levinson and Redheffer (1970).

Return to Complex Analysis.

If $ C $ is a closed contour (Figure 1.), and the complex valued function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(z) is an analytic function of the complex variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z inside the region bounded by, and on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \int_C f(z) \; dz = 0 .

Proof

If we substitute for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(z) = u(x,y) + i v(x,y) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): dz = dx + i dy

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \int_C f(z) \; dz = \int_C (u(x,y) + i v(x,y)) \; (dx + i dy) = \int_C u(x,y) \; dx - \int_C v(x,y) \; dy + i \left( \int_C u(x,y) \; dy + \int_C v(x,y) \; dx \right).

By Green's theorem, for any two functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): P(x,y) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q (x,y) such that $ {\frac {\partial P}{\partial y}} $ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \frac{\partial Q}{\partial x } exist in a two dimensional region Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathcal R} bounded by a curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \int_{\mathcal R} \int \left(\frac{\partial Q}{\partial x } - \frac{\partial P}{\partial y}\right) \; dx \; dy = \int_C P \; dx + \int_C Q \; dy .

If we apply Green's theorem to the real and complex terms of the integral above, we have, identifying the real and imaginary parts of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(x) with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): P and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q where appropriate, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \int_C f(z) \; dz = - \left( \int_{\mathcal R} \int \left(\frac{\partial v(x,y)}{\partial x } + \frac{\partial u(x,y) }{\partial y}\right) \; dx \; dy \right) + i \left( \int_{\mathcal R} \int \left(\frac{\partial u(x,y)}{\partial x } - \frac{\partial v(x,y)}{\partial y}\right) \; dx \; dy \right).

Because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(z) is analytic inside Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathcal R} its real and imaginary parts must satisfy the Cauchy-Riemann equations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \frac{\partial u(x,y)}{\partial x } = \frac{\partial v(x,y)}{\partial y} \quad \quad \mbox{and} \quad \quad \frac{\partial v(x,y)}{\partial x } = -\frac{\partial u(x,y)}{\partial y}.

Thus the real and imaginary parts vanish independently showing that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \int_C f(z) \; dz = 0.

We note that the shape of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C is quite general. It may have any shape, as long as it does not cross itself, and may have any finite number of corners, where the function describing the curve is continuous, but not differentiable.

The extension of Cauchy's theorem to a region with any finite number of holes is called the Cauchy-Goursat theorem.

Cauchy Goursat theorem

"Figure 2: f(z) is analytic in the shaded region"

If a complex valued function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(z) is analytic in a region of the complex plane bounded by a simple closed curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_1 , except possibly on any number of finite subdomains (holes) bounded by simple closed curves Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_k for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k> 1 then Cauchy's theorem holds in that region bounded by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_1 and all of the curves Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_k .

Proof:

Consider a region bounded by a simple closed curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_1 with a hole bounded by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_2 . (See Figure 2.) We may connect the two regions with a cut long the curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): [a,b]. The integral over the full boundary of the shaded region, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(z) is analytic is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \int_{C_1 + [a,b] + C_2 + [b,a]} f(z) \; dz = \left\{ \oint_{C_1} + \int_a^b + \oint_{C_2^{-}} + \int_b^a \right\} f(z) \; dz = 0

where the notation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_2^{-} indicates that the integration path is in the clockwise (negative) direction in the complex plane.

"Figure 3: equivalent contour integrals"

Because

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \int_a^b f(z) \; dz = - \int_b^a f(z) \; dz

we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \int_{C_1} f(z) \; dz = \int_{C_2^{-}} f(z) \; dz

Reversing the direction of integration on the integral on the right hand side yields

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \int_{C_1} f(z) \; dz = \int_{C_2} f(z) \; dz.

Thus the integrals over the integration contours Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_1 and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_2 are equivalent. Because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(z) need not be analytic in the interior of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_2 these integrals are not necessarily zero.

We may have any finite number of holes in our domain, and the sum of the integrals over the curves Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_k bounding these holes is equivalent to the integral over the bounding contour Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_1. (See Figure 3.)

Another way of interpreting this result is that we may continuously deform the countour Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_1 to any other closed simple curve $ C_{2} $ enclosing the same region. Again, there is no restriction on the shape of the contours, only that they are connected, and that they have at most a finite number of corners.

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