Solving Complex Analysis Questions: Are My Answers Right?

In summary: It's actually \oint \frac{\cos(z+11)}{z^2-23} dzand since z = ±√23 is not in the unit circle, it must be 0 right?The second integral is \oint \frac{\cos(z+11)}{z^2-6z} dzwhich can be written as \oint \frac{\cos(z+11)}{z(z-6)} dzthen \oint \frac{\frac{\cos(z+11)}{z-6}}{z} dzso evaluating \oint \frac{\
  • #1
mick25
13
0

Homework Statement



I just wrote a test and was wondering if I got these questions right, I already solved them, please see the attached pictures below. Here are the questions; sorry for non-latex form

1) Let gamma be a positively oriented unit circle (|z|=1) in C

solve: i) integral of cos(z+11)/(z^2-23) along gamma
ii) integral of cos(z+11)/(z^2-6z) along gamma

2) Find all possible analytic functions f(z) such that |sin 7z| < |f(z)|

3) Draw a closed curve gamma in C\<0,1> such that its winding numbers at 0 is 1 and at 1 is -2.

4) Draw 4 pairwise non-homotopic closed curves in C\<0>

The Attempt at a Solution



Here are the pictures:

33p6k3d.jpg

i5wzeq.jpg

2qbvcer.jpg


Are these correct? Please let me know if my writing is ineligible. Thanks in advance.
 
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  • #2


Your writing is illegible to me. Not too hard to do latex. Let's see if I can write the first one:

[tex]\oint \frac{\cos(z+11)}{z^2+23} dz,\quad z=e^{it}[/tex]

Looks all analytic in there so that's zero right? Think you said that. Now if you want, do a quote on my post to see the Latex code and if you want to post more in here, try and learn how to do it.
 
  • #3


jackmell said:
Your writing is illegible to me. Not too hard to do latex. Let's see if I can write the first one:

[tex]\oint \frac{\cos(z+11)}{z^2+23} dz,\quad z=e^{it}[/tex]

Looks all analytic in there so that's zero right? Think you said that. Now if you want, do a quote on my post to see the Latex code and if you want to post more in here, try and learn how to do it.

It's actually

[tex]\oint \frac{\cos(z+11)}{z^2-23} dz[/tex]

and since [tex] z = ±√23[/tex] is not in the unit circle, it must be 0 right?

The second integral is

[tex]\oint \frac{\cos(z+11)}{z^2-6z} dz[/tex]

which can be written as

[tex]\oint \frac{\cos(z+11)}{z(z-6)} dz[/tex]

then

[tex]\oint \frac{\frac{\cos(z+11)}{z-6}}{z} dz[/tex]

so evaluating [tex]\oint \frac{\cos(z+11)}{z-6} dz at z=0[/tex] gives [tex]\frac{pi*icos(11)}{3}[/tex]
 
  • #4


mick25 said:
It's actually

[tex]\oint \frac{\cos(z+11)}{z^2-23} dz[/tex]

and since [tex] z = ±√23[/tex] is not in the unit circle, it must be 0 right?

The second integral is

[tex]\oint \frac{\cos(z+11)}{z^2-6z} dz[/tex]

which can be written as

[tex]\oint \frac{\cos(z+11)}{z(z-6)} dz[/tex]

then

[tex]\oint \frac{\frac{\cos(z+11)}{z-6}}{z} dz[/tex]

so evaluating [tex]\oint \frac{\cos(z+11)}{z-6} dz at z=0[/tex] gives [tex]\frac{pi*icos(11)}{3}[/tex]

That's very good except the last one I get [itex]-\frac{\pi i\cos(11)}{3}[/itex] Also, you say "since [itex]z=\pm \sqrt{23}[/itex], really should say "the poles are at zero and [itex]z=\pm \sqrt{23}[/itex]". What else? Yeah, that "dzat" thing. Kinda' problematic to include text and math code in a single line. Should have just put the integral, then say "at z" right below it. And the pi. The math code for that is \pi. Also, now that you posted that big notebook paper picture (any large pictures) it has skewed the entire thread so you can't view it without scrolling. That's a pain. Why not just remove it so it's easier to see the thread, then ask more question about it just using latex. Won't back here till tomorrow though. Others may help though.
 

Related to Solving Complex Analysis Questions: Are My Answers Right?

1. How do I know if my answers are correct?

The best way to check if your answers are correct is to compare them with the solutions provided in your textbook or by your instructor. If your answers match the solutions, then they are most likely correct. You can also ask for feedback from your classmates or instructor.

2. What should I do if I am not sure about my answers?

If you are not sure about your answers, you can try solving the problem again using a different approach or method. You can also consult your textbook or notes for any examples or similar problems that can help you verify your answers. Additionally, you can seek help from your classmates or instructor.

3. How can I improve my problem-solving skills in complex analysis?

The best way to improve your problem-solving skills in complex analysis is to practice regularly. Try solving different types of problems and challenge yourself with more difficult ones. You can also attend review sessions or join study groups to learn from your peers and discuss different problem-solving strategies.

4. What are some common mistakes to avoid when solving complex analysis questions?

Some common mistakes to avoid when solving complex analysis questions include not reading the question carefully, not understanding the concepts and definitions, and not showing all the steps in your solution. It is also important to check your calculations and make sure you are using the correct formulas and theorems.

5. Is it necessary to show all the steps in my solution?

Yes, it is important to show all the steps in your solution in complex analysis. This allows you to identify any mistakes and helps your instructor understand your thought process. Additionally, showing your work can earn you partial credit even if your final answer is incorrect.

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