1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

7 .5 • APPLICAT IONS OF TAYLOR AND L AURENT SERIES 287

Solution Using long division, we see that

1 1 12 54
secz = --= 2 • • = 1 + - 2 z + 24 z + · · · ·
cosz 1 - ~ + ~ -ii-+ ...

Moreover, using Taylor's theorem, we see that if f (z) = secz, then J<~/^0 > = { 4 ,

so f <^4 > (O) = 5.

We close this section with some results concerning the behavior of complex
functions at points near the different types of isolated singularities. Theorem 7 .15
is due to the German mathematician G. F. Bernhard Riemann (1826-1866).

The proof of Corollary 7.12 is given as an exercise.