Exercise · ZW-COMPLEX-TAYLOR-SERIES-001 · ★☆☆ Basic
The coefficient of \(z^3\) in the Taylor series of \(e^z\) at \(z_0=0\) is:
Choose one option.
Show answer
- B. \(\tfrac16\)
Explanation
\(a_3=f'''(0)/3!=1/6\), since all derivatives of \(e^z\) equal \(1\) at \(0\).
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
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- ZW-COMPLEX-TAYLOR-SERIES-009 The coefficient of z^4 in the Taylor series of z^2e^z at 0 is: