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Date: 27-11-2018
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Cauchy's integral formula states that
(1) |
where the integral is a contour integral along the contour enclosing the point .
It can be derived by considering the contour integral
(2) |
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
(3) |
so
(4) |
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
(5) |
Now, let , so . Then
(6) |
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(7) |
But we are free to allow the radius to shrink to 0, so
(8) |
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(9) |
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(10) |
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(11) |
giving (1).
If multiple loops are made around the point , then equation (11) becomes
(12) |
where is the contour winding number.
A similar formula holds for the derivatives of ,
(13) |
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(14) |
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(15) |
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(16) |
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(17) |
Iterating again,
(18) |
Continuing the process and adding the contour winding number ,
(19) |
REFERENCES:
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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