Exercise · ZW-COMPLEX-TAYLOR-SERIES-004 · ★★☆ Intermediate
What is the radius of convergence of the Taylor series of \(\dfrac1{z^2+1}\) at \(z_0=0\)?
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- C. \(1\)
Explanation
The singularities are \(\pm i\), at distance \(1\) from \(0\). Although the function is smooth on \(\mathbb{R}\), the radius is \(1\).
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-005 What if the centre is z_0=2? The radius for 1z^2+1 is:
- ZW-COMPLEX-TAYLOR-SERIES-006 The coefficient of (z-1)^3 in the Taylor series of 1/z at z_0=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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