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


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




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),


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, Pole

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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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Cauchy Integral Formula

Cite this as:

Weisstein, Eric W. "Cauchy Integral Formula." From MathWorld--A Wolfram Web Resource.

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