Cauchy Integral Formula
Cauchy's integral formula states that
where the integral is a contour integral along the contour gamma enclosing the point z_0.
It can be derived by considering the contour integral
| [画像: ∮_gamma(f(z)dz)/(z-z_0), ] |
(2)
|
defining a path gamma_r as an infinitesimal counterclockwise circle around the point z_0, and defining the path gamma_0 as an arbitrary loop with a cut line (on which the forward and reverse contributions cancel each other out) so as to go around z_0. The total path is then
| gamma=gamma_0+gamma_r, |
(3)
|
so
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 gamma_0 does not enclose the pole, and we are left with
Now, let z=z_0+re^(itheta), so dz=ire^(itheta)dtheta. Then
But we are free to allow the radius r to shrink to 0, so
giving (1).
If multiple loops are made around the point z_0, then equation (11) becomes
where n(gamma,z_0) is the contour winding number.
A similar formula holds for the derivatives of f(z),
![画像:lim_(h->0)1/(2piih)[∮_gamma(f(z)dz)/(z-z_0-h)-∮_gamma(f(z)dz)/(z-z_0)]](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/CauchyIntegralFormula/Inline37.svg){kind=link}
![画像:lim_(h->0)1/(2piih)∮_gamma(f(z)[(z-z_0)-(z-z_0-h)]dz)/((z-z_0-h)(z-z_0))](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/CauchyIntegralFormula/Inline40.svg){kind=link}
Iterating again,
Continuing the process and adding the contour winding number n,
See also
Argument Principle, Cauchy Integral Theorem, Complex Residue, Contour Integral, Morera's Theorem, PoleExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Cauchy Integral FormulaCite this as:
Weisstein, Eric W. "Cauchy Integral Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CauchyIntegralFormula.html