Solving Cauchy Residual Theorem for Holomorphic Function at z=2i

[Alekon]

Alright so I posted a picture asking the exact question.

Here is my best attempt...

According to my professor's terrible notes, the numerator can magically turn into the form:

e^i(z+3)

when converted to complex. The denominator will be factored into

(z-2i)(z+2i)

but the function is only holomorphic at z=2i so only (z+2i) can be used.

From there the Res(f,2i)=g(2i) which is equal to what I believe is something like

e^(i(2i+3)/(4i)

It follows that

J=e^(-2+3i)*Pi

and sovling for the real part gives me an incorrect answer.

I might be missing some steps but I'm going off a theorem and it's really hard to relate to this problem. Help me PLEASE!

 

[Herick]

Is it 0.00477463?

 

[Herick]

This is $$\frac{\sin (3)}{4 e^2}$$

Let me know if this is the answer. I can explain how i got it.

 

[Alekon]

I finally calculated the answer... it turned out to be -0.2

See the picture if you're interested

 

[blue_raver22]

I can understand your frustration with this problem. The Cauchy Residual Theorem is a powerful tool for solving complex integrals, but it can be tricky to apply correctly. Let's break down the steps and see if we can find where things might have gone wrong.

First, we need to understand what the Cauchy Residual Theorem is saying. It states that for a holomorphic function f(z) and a point z0 inside a closed contour C, the value of the integral of f(z) around C can be calculated by using the residue of f(z) at z0. In other words, the integral is equal to 2πi times the residue of f(z) at z0.

Now, in your problem, the function f(z) is not given, so we need to find it first. From the given information, we know that the function is only holomorphic at z=2i. This means that f(z) must have a singularity at z=2i, and we can use the Cauchy Residual Theorem to find its residue at that point.

Next, we need to factor the denominator of f(z) into (z-2i)(z+2i). However, since the function is only holomorphic at z=2i, we can only use the term (z+2i) in our calculation. This means that the residue of f(z) at z=2i is given by the formula Res(f,2i)=lim(z->2i) (z+2i)f(z).

From here, we can substitute z=2i into the function and simplify to find the residue of f(z) at z=2i. Finally, we can use the Cauchy Residual Theorem to calculate the value of the integral of f(z) around the given contour.

It's possible that your answer is incorrect because of a mistake in the calculation of the residue or in the substitution of z=2i into the function. Double check your work and make sure you're following the steps correctly. If you're still having trouble, I recommend seeking help from your professor or a tutor who can guide you through the process step by step. Keep practicing and don't get discouraged, solving complex problems like this takes time and practice. Good luck!