Exercise · ZW-COMPLEX-TAYLOR-SERIES-006 · ★★☆ Intermediate
The coefficient of \((z-1)^3\) in the Taylor series of \(1/z\) at \(z_0=1\) is:
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- C. \(-1\)
Explanation
\(\tfrac1z=\tfrac1{1+(z-1)}=\sum(-1)^n(z-1)^n\), so \(a_3=-1\).
Review the concept: Complex Taylor series
The Taylor series of a function \(f\) holomorphic on a disc \(|z-z_0|<R\) is the power series in \(z-z_0\), centred at \(z_0\), whose coefficients \(a_n\) come from the derivatives of \(f\) at \(z_0\).
\[f(z)=\sum_{n=0}^{\infty}a_n(z-z_0)^n,\quad a_n=\frac{f^{(n)}(z_0)}{n!}\]
More practice problems: complex taylor series
- ZW-COMPLEX-TAYLOR-SERIES-001 The coefficient of z^3 in the Taylor series of e^z at z_0=0 is:
- ZW-COMPLEX-TAYLOR-SERIES-002 The coefficient of z^5 in the Taylor series of sin z at 0 is:
- ZW-COMPLEX-TAYLOR-SERIES-003 The radius of convergence of the Taylor series of 11-z at 0 is:
- ZW-COMPLEX-TAYLOR-SERIES-004 What is the radius of convergence of the Taylor series of 1z^2+1 at z_0=0?
- ZW-COMPLEX-TAYLOR-SERIES-005 What if the centre is z_0=2? The radius for 1z^2+1 is:
- ZW-COMPLEX-TAYLOR-SERIES-007 The radius of convergence of the Taylor series of tan z at 0 is:
- ZW-COMPLEX-TAYLOR-SERIES-008 If f is entire, its Taylor series at 0 converges to f(z):
- ZW-COMPLEX-TAYLOR-SERIES-009 The coefficient of z^4 in the Taylor series of z^2e^z at 0 is:
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