Complex Analysis: Theory and Applications · Problem 7.2.2

Taylor and Laurent Series — Howell & Matthews

Mathematics — Complex Analysis · taylor-series

Problem

Using methods other than computing derivatives, find the Maclaurin series for ((a))

\cos^3 z

. Hint: Use the trigonometric identity

4\cos^3 z = \cos 3z + 3\cos z

. ((b))

\mathrm{Arctan}\, z

. Hint: Choose an appropriate contour and integrate the second series in Equations (7.12). ((c))

f(z) = (z^2+1)\sin z

. ((d))

f(z) = e^z \cos z

. Hint:

\cos z = \frac{1}{2}(e^{iz}+e^{-iz})

, so

f(z) = \frac{1}{2}e^{(1+i)z} + \frac{1}{2}e^{(1-i)z}

. Now use the Maclaurin series for

e^z