Cauchy's integral formula states that
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(1)
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where the integral is a contour integral along the contour enclosing the point .
It can be derived by considering the contour
integral
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(2)
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defining a path as an infinitesimal
counterclockwise circle around the point
, and defining the path as an arbitrary
loop with a cut line (on which the forward and reverse contributions cancel each
other out) so as to go around . The total path
is then
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(3)
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so
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(4)
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From the Cauchy integral theorem, the contour integral
along any path not enclosing a pole is
0. Therefore, the first term in the above equation is 0 since does not
enclose the pole, and we are left with
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(5)
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Now, let , so . Then
But we are free to allow the radius to shrink to 0,
so
giving (1).
If multiple loops are made around the point , then equation
(10) becomes
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(11)
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where is the contour winding number.
A similar formula holds for the derivatives of ,
Iterating again,
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(17)
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Continuing the process and adding the contour winding number ,
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(18)
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Arfken, G. "Cauchy's Integral Formula." §6.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL:
Academic Press, pp. 371-376, 1985.
Kaplan, W. "Cauchy's Integral Formula." §9.9 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley,
pp. 598-599, 1991.
Knopp, K. "Cauchy's Integral Formulas." Ch. 5 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part
I. New York: Dover, pp. 61-66, 1996.
Krantz, S. G. "The Cauchy Integral Theorem and Formula." §2.3 in Handbook
of Complex Variables. Boston, MA: Birkhäuser, pp. 26-29, 1999.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill,
pp. 367-372, 1953.
Woods, F. S. "Cauchy's Theorem." §146 in Advanced Calculus: A Course Arranged with Special Reference to
the Needs of Students of Applied Mathematics. Boston, MA: Ginn, pp. 352-353,
1926.
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