Cauchy's theorem
Objective. Use Cauchy's theorem and integral formula.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Cauchy's theorem states that if f is holomorphic throughout a simply connected domain D, then ∮_C f(z) dz = 0 for every closed contour C in D, since there are no singularities to obstruct path-independence. Building on this, Cauchy's integral formula gives f(a) = (1/2πi)∮_C f(z)/(z−a) dz for a inside C, allowing function values to be recovered from boundary data alone. These results explain why contour integrals of 1/z around loops enclosing the origin give 2πi, since 1/z is not holomorphic at z = 0, the one singular point inside the loop. Together they underpin residue calculus for evaluating integrals.
Worked example
Use Cauchy's integral formula to evaluate ∮_C (e^z)/(z−1) dz where C is the circle |z| = 2.
- 1Identify f(z) = e^z, which is holomorphic everywhere, and a = 1 lies inside |z| = 2.
- 2Cauchy's integral formula gives ∮_C f(z)/(z−a) dz = 2πi·f(a).
- 3Compute f(1) = e^1 = e.
- 4So the integral equals 2πi·e.
Answer: 2πi·e
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Practice
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Start practisingCommon mistake. Students apply Cauchy's theorem to conclude an integral is zero without checking that the function is holomorphic everywhere inside the contour; a pole inside the loop, as with f(z) = 1/z at the origin, invalidates the zero conclusion.
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