Cauchy Integral formulas

The Cauchy Integral formulas for 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) which is analytic inside and on 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 in some region $ {\mathcal {R}} $ of the complex 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 plane, for a complex number 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 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/":): 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/":): f^{(n)} (a) = \frac{n!}{2 \pi i } \int_{C} \frac{f(z)}{(z - a)^{n+1} } dz.

Proof by induction:

Proof of the 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/":): n =1 case

We consider the case 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/":): n =1 . We begin by using the Cauchy integral formula to formally define the derivative 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(z)

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/":): \lim_{h \rightarrow 0} \frac{ f( a + h) - f(a) }{h} = \lim_{h \rightarrow 0} \frac{1}{2 \pi i } \int_{C}f (z)\frac{1}{h} \left[ \frac{1}{(z - a - h)} - \frac{1}{z - a} \right] \; dz

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/":): = \lim_{h \rightarrow 0} \frac{1}{2 \pi i } \int_{C}f (z) \frac{1}{h}\left[ \frac{z - a - (z - a -h )}{(z -a )(z -a - h) } \right] \; dz = \lim_{h \rightarrow 0} \frac{1}{2 \pi i } \int_{C}f (z) \left[ \frac{1}{(z-a-h)(z-a) } \right] \; dz

Multiplying and dividing 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/":): (z -a) and adding and subtracting <mmath> h </math> in the numerator allows us to write

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/":): = \lim_{h \rightarrow 0} \frac{1}{2 \pi i } \int_{C}f (z) \left[ \frac{z -a -h + h}{(z-a-h)(z-a)^2 } \right] \; dz = \frac{1}{2 \pi i } \int_{C} \frac{f(z)}{(z-a)^2 } \; dz + \lim_{h \rightarrow 0} \frac{1}{2 \pi i } \int_{C}f (z) \left[ \frac{h}{(z-a-h)(z-a)^2 } \right] \; dz.

We may estimate the second integral noting 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/":): | z - a - h | \ge |z - a | - |h| and considering deforming the contour of integration to a circle of radius 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/":): R such 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/":): C: |z - a| = R and from the maximum modulus theorem 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/":): |f(z)| \le M

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/":): \lim_{h \rightarrow 0} \left| \frac{1}{2 \pi i } \oint_C \frac{ f(z) }{(z - a - h) (z -a)^2 } \; dz \right| < \lim_{h \rightarrow 0}\frac{|h|}{2 \pi} \frac{2 \pi |z - a| M }{|| z -a | - |h| | |z -a|^2} \le\lim_{h \rightarrow 0} \frac{M|h|}{\left| 1 - \displaystyle \frac{h}{R} \right| R^2} = 0,

proving the theorem for the case 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/":): n=1 .

It is possible to prove a finite number of cases using a similar construction.

Proof going from 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 to the 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 -th case

For the 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/":): n=k -th case we can consider beginning with the $ n=k-1 $ case and formally constructing the derivative

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^{(k)} (a) = \lim_{w \rightarrow a} \frac{f^{(k-1)} ( w ) - f^{(k-1)}(a)}{ w - a } = \lim_{w \rightarrow a }\frac{(k-1)!}{2 \pi i} \oint_C f(z) \frac{1}{w - a } \left[\frac{1}{(z - w)^k} - \frac{1}{(z-a)^k } \right] \; dz

In the limit,

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/":): \lim_{w \rightarrow a} \frac{ \displaystyle \frac{1}{z -w } - \displaystyle \frac{1}{z -a } }{w -a} = \frac{\partial}{\partial a } \left[\frac{ 1 }{(z - a )^{k} }\right] = \frac{k}{(z -a)^{k+1}}.

Thus,

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^{(k)} (a) = \frac{k!}{2 \pi i } \oint_C \frac{f(z)}{(z -a )^{k+1} } \; dz .

We note that it is also prove the general case by differentiating each side of the Cauchy integral formula 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/":): n times with respect to 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 , where the 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/":): n -th partial derivative with respect to 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 is brought inside the integral

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{d^n f(a)}{d a^n} = \frac{1}{2 \pi i } \oint_C f(z) \frac{\partial^n}{\partial a^n}\left[ \frac{1}{(z - a)} \right]\; dz = \frac{n!}{2 \pi i} \oint_C \frac{f(z)}{(z -a)^{n+1}} \; dz .

This is the complex integral form of the Leibniz rule for differentiating under the integral sign.

Return to Complex Analysis.