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Poopsilon
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Edit: Never mind I found my error, moderator can lock this.
Evaluate the integral \int_0^{\pi} \frac{dt}{(a+cost)^2} for a > 1.
\int_0^{\pi}\frac{dt}{(a+cost)^2} = \pi i\sum_{a\epsilon \mathbb{E}}Res(f;\alpha)
Where \mathbb{E} is the open unit disk, and f(z) = \frac{1}{iz(a+\frac{1}{2}(z+\frac{1}{z}))^2}.
f(z) = \frac{1}{iz(a+\frac{1}{2}(z+\frac{1}{z}))^2} = \frac{1}{iz\frac{(z^2 + 2az + 1)^2}{4z^2}} = \frac{-4iz}{(z^2 + 2az + 1)^2} = \frac{-4iz}{[(z + a + \sqrt{a^2 - 1})(z + a - \sqrt{a^2 - 1})]^2}
Thus f has two roots of multiplicity two. It's fairly easy to see that only the root z = -a + \sqrt{a^2 - 1} lies inside the open unit disk for a > 1, thus we set b = -a + \sqrt{a^2 - 1} and obtain:
\pi i\sum_{\alpha\epsilon\mathbb{E}}Res(f;\alpha) = Res(f;b) = lim_{z\rightarrow b}\frac{d}{dz}\frac{-4iz(z+a-\sqrt{a^2 - 1})^2}{(z + a + \sqrt{a^2 - 1})^2(z + a - \sqrt{a^2 - 1})^2} = lim_{z\rightarrow b}4\pi\frac{d}{dz}\frac{z}{(z+a+\sqrt{a^2 - 1})^2} = lim_{z\rightarrow b}4\pi \frac{a + \sqrt{a^2 - 1} -z}{(z + a + \sqrt{a^2 - 1})^3}
From here I take the limit and after simplification obtain \frac{\pi\sqrt{a^2 - 1}}{a^3}. Yet the book says my answer should be \frac{\pi a}{(a^2 - 1)\sqrt{a^2 - 1}}. Where have I gone wrong? Thanks.
Homework Statement
Evaluate the integral \int_0^{\pi} \frac{dt}{(a+cost)^2} for a > 1.
Homework Equations
\int_0^{\pi}\frac{dt}{(a+cost)^2} = \pi i\sum_{a\epsilon \mathbb{E}}Res(f;\alpha)
Where \mathbb{E} is the open unit disk, and f(z) = \frac{1}{iz(a+\frac{1}{2}(z+\frac{1}{z}))^2}.
The Attempt at a Solution
f(z) = \frac{1}{iz(a+\frac{1}{2}(z+\frac{1}{z}))^2} = \frac{1}{iz\frac{(z^2 + 2az + 1)^2}{4z^2}} = \frac{-4iz}{(z^2 + 2az + 1)^2} = \frac{-4iz}{[(z + a + \sqrt{a^2 - 1})(z + a - \sqrt{a^2 - 1})]^2}
Thus f has two roots of multiplicity two. It's fairly easy to see that only the root z = -a + \sqrt{a^2 - 1} lies inside the open unit disk for a > 1, thus we set b = -a + \sqrt{a^2 - 1} and obtain:
\pi i\sum_{\alpha\epsilon\mathbb{E}}Res(f;\alpha) = Res(f;b) = lim_{z\rightarrow b}\frac{d}{dz}\frac{-4iz(z+a-\sqrt{a^2 - 1})^2}{(z + a + \sqrt{a^2 - 1})^2(z + a - \sqrt{a^2 - 1})^2} = lim_{z\rightarrow b}4\pi\frac{d}{dz}\frac{z}{(z+a+\sqrt{a^2 - 1})^2} = lim_{z\rightarrow b}4\pi \frac{a + \sqrt{a^2 - 1} -z}{(z + a + \sqrt{a^2 - 1})^3}
From here I take the limit and after simplification obtain \frac{\pi\sqrt{a^2 - 1}}{a^3}. Yet the book says my answer should be \frac{\pi a}{(a^2 - 1)\sqrt{a^2 - 1}}. Where have I gone wrong? Thanks.
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