Cauchy Integral Formula
المؤلف:
Krantz, S. G.
المصدر:
"The Cauchy Integral Theorem and Formula." §2.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser
الجزء والصفحة:
...
17-11-2018
2250
Cauchy Integral Formula
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
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 (11) becomes
 |
(12)
|
where 
is the contour winding number.
A similar formula holds for the derivatives of 
,
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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