Simple complex integral 2

1. Oct 18, 2007

John O' Meara

write f(z) in terms of partial fractions and integrate it counterclockwise over the unit circle where $$f(z)=\frac{2z+3\iota}{z^2+\frac{1}{4}} \\$$
$$2\int\frac{z}{z^2 + \frac{1}{4}} dz \ + \ 3\iota \int\frac{1}{z^2 + \frac{1}{4}}dz \\ \ \mbox{. Also }$$
$$\ z(t) = \exp{\iota t} (0 \leq \ t \ \leq2\pi) \ \frac{dz}{dt}=\iota \exp{\iota t} \\ \ \mbox{therefore} \ f(z(t))=\frac{\exp{\iota t}}{\exp{2\iota t} + \frac{1}{4}}\\ \$$
$$\ 2\iota \int_0 ^{2\pi} \frac{\exp{2\iota t}}{\exp{2\iota t} + \frac{1}{4}}dt \\$$ I let $$u=\exp{2\iota t} + \frac{1}{4} \\ \ \mbox{ therefore I get } \ \\ \ \int_1 ^1 \frac{1}{u} du=0\\$$. I also get 0 for the 2nd integral expression. Please check my results for error. Thanks.
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