Exercise · ZW-CAUCHY-INTEGRAL-FORMULA-004 · ★☆☆ Basic
If \(f\) is holomorphic on the closed disk \(|z|\le 1\) and \(\oint_{|z|=1}\frac{f(z)}{z}\,dz=4\pi i\), what is \(f(0)\)?
Choose one option.
Show answer
- B. \(2\)
Explanation
The formula gives \(\oint\frac{f(z)}{z}dz=2\pi i\,f(0)\). So \(2\pi i\,f(0)=4\pi i\) and \(f(0)=2\).
Review the concept: Cauchy integral formula
The Cauchy integral formula gives the value of a holomorphic function \(f\) at a point \(z_0\) inside a simple closed contour \(C\) by means of an integral of \(f\) over \(C\).
\[f(z_0)=\frac{1}{2\pi i}\oint_C\frac{f(z)}{z-z_0}\,dz\]
More practice problems: cauchy integral formula
- ZW-CAUCHY-INTEGRAL-FORMULA-001 Compute oint_|z|=2(e^z)/(z-1) dz, with the circle traversed counterclockwise.
- ZW-CAUCHY-INTEGRAL-FORMULA-002 Compute oint_|z|=1(cos z)/(z) dz counterclockwise.
- ZW-CAUCHY-INTEGRAL-FORMULA-003 Compute oint_|z|=3(z^2)/(z-i) dz counterclockwise.
- ZW-CAUCHY-INTEGRAL-FORMULA-005 Compute oint_|z|=4(sin z)/(z-π) dz counterclockwise.
- ZW-CAUCHY-INTEGRAL-FORMULA-006 Compute oint_|z|=1(e^z)/(z-3) dz counterclockwise.
- ZW-CAUCHY-INTEGRAL-FORMULA-007 Compute oint_|z|=1(e^z)/(z) dz traversing the circle clockwise.
- ZW-CAUCHY-INTEGRAL-FORMULA-008 Compute oint_|z|=3(z)/(z^2-1) dz counterclockwise.
- ZW-CAUCHY-INTEGRAL-FORMULA-009 Which hypothesis is needed to apply the Cauchy integral formula to oint_C(f(z))/(z-z_0)dz?
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