# Complex Analysis - Residues

iflare
Hello!

I am studing for my Complex Analysis exam and solving the exercises for Residues given by the professor.

The problem is that for some exercises I get to a solution different from the one of the professor , and I am not sure that the mistake is in my calculations.

I would greatly appreciate it, if somebody could solve it and tell me what a solution he/she came up with.

Here is the exercise:
Calculate the residue of the complex-valued function $$f(z)$$ at $$z=-\imath$$, as:

$$f(z) = \frac{\sin(z)}{(z^2 + 1)^2}$$​

$$Res(f(z),\imath) = -\frac{1}{4e}$$​
$$Res(f(z),\imath) = \frac{\imath}{2}\cosh(1)$$​

Thanks a lot!

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Staff Emeritus
Well, what did you get? Post your attempt, and we'll be able to see if you went wrong.

iflare
Well, what did you get? Post your attempt, and we'll be able to see if you went wrong.

Thank you for getting involved , here is what I did, I used the formula:

$$Res(f(z), z_0) = \lim_{z \to z_0} \left( \frac{1}{(m-1)!}\, \dfrac{d^{m-1}}{dz^{m-1}} \left( (z-z_0)^m f(z) \right) \right)$$​

where $$m$$ is the order of the pole, and $$z_0$$ is the pole.

In the particular case of this exercise,

\begin{align*} m &= 2 \\ z_0 &= -i \end{align*}

\begin{align*} Res(f(z), -i) &= \lim_{z \to z_0} \left( \frac{1}{(m-1)!}\, \dfrac{d^{m-1}}{dz^{m-1}} \left( (z-z_0)^m f(z) \right) \right) \\ &= \lim_{z \to -i} \left( \frac{1}{1!} \, \dfrac{d}{dz} \left( (z+i)^2 f(z) \right) \right) \end{align*}

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iflare
My Calculations

Here are the calculations:

\begin{align*} Res(f(z), -i) &= \lim_{z \to z_0} \left( \frac{1}{(m-1)!} \, \dfrac{d^{m-1}}{dz^{m-1}} \left( (z-z_0)^m f(z) \right) \right) \\ &= \lim_{z \to -i} \left( \frac{1}{1!} \, \dfrac{d}{dz} \left( (z+i)^2 f(z) \right) \right) \\ &= \lim_{z \to -i} \left( \dfrac{d}{dz} \left( (z+i)^2 \frac{\sin(z)}{(z+i)^2 (z-i)^2} \right) \right) \\ &= \lim_{z \to -i} \left( \dfrac{d}{dz} \left(\frac{\sin(z)}{(z-i)^2} \right) \right) \\ &= \lim_{z \to -i} \left( \dfrac{\cos(z) \cdot (z-i)^2 - \sin(z) \cdot 2(z-i)(z-i)'}{(z-i)^4} \right) \\ &= \dfrac{\cos(-i) \cdot (-i-i)^2 - \sin(-i) \cdot 2(-i-i)}{(-i-i)^4} = \dfrac{\cos(-i) \cdot 4(-1) +4i \sin(-i)}{(-2i)^4} \\ &= \dfrac{-4\cos(-i) +4i \sin(-i)}{16} = \dfrac{-\cos(-i) +i \sin(-i)}{4} = \dfrac{-\cos(i) -i \sin(i)}{4}\\ &= -\dfrac{\cos(i) + i \sin(i)}{4} = -\dfrac{e^{i \cdot i}}{4}= -\dfrac{e^{-1}}{4} = -\dfrac{1}{4e}\\ \end{align*}

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iflare
The Professor's Solution

Here is the solution of the professor. This is a screenshot of the page of his lecture notes on which he solves the exercise. In his calculations:
$$j=\sqrt{-1}$$​

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Homework Helper
iflare, the fact of the matter is that your prof goofed the derivative and your work is correct.

iflare
iflare, the fact of the matter is that your prof goofed the derivative and your work is correct.

That is just great , thank you very much for the help!

Gold Member
i don't think it's so great that the prof goofed here!

Homework Helper
i haven't taught this in a long time, so this may be wrong, but i think the idea is to separate the pole from the holomorphic part at -i.

so we have sin(z)/(z-i)^2 as the holomorphic part and 1/(z+i)^2 as the polar part.

now we want to multiply these together and pick off the coefficient of
1/(z+i)

Now to get the coefficient of 1/(z+i) it seems we just need the derivative of the holo part at -i.

by the quotient rule that should be [cos(z)(z-i) - 2sin(z)]/(z-i)^3, all evaluated at z= -i. yipes!

i.e. -[cos(i)+i sin(i)]/4 = -1/4e,

using the fact that cos(z) = (1/2)[e^z + e^(-z)], etc...

so its much easier than it looks above, but still tedious for me.

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