Exercise · ZW-CAUCHY-INTEGRAL-FORMULA-001 · ★☆☆ Basic
Compute \(\displaystyle\oint_{|z|=2}\frac{e^z}{z-1}\,dz\), with the circle traversed counterclockwise.
Choose one option.
Show answer
- C. \(2\pi i\,e\)
Explanation
The point \(z_0=1\) lies inside \(|z|=2\) and \(e^z\) is entire, so the integral equals \(2\pi i\,f(1)=2\pi i\,e\).
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-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-004 If f is holomorphic on the closed disk |z|≤ 1 and oint_|z|=1(f(z))/(z) dz=4π i, what is f(0)?
- 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?