Verification of contour integral and residue theorem

[shrewd bay]

I split the contour into parts, since residue theorem requires a simple contours, which this one is not yet. I seperated all little loops. The integrals for the little loops are zero except for the one around the singularity -1. The big loop goes around singularities -1, 0, 1 (where 0 is a removable singularity). Residue theorem then tells us that the integral of the big loop evaluates to pi i (sin(-1)+sin(1)) = 0.
So only the little loop around -1 determines the total integral and by residue theorem again, this evaluates to -pi i sin(1).
Is this correct? Was there a quicker way to see that the big loop integral would be 0?
Thanks in advance!

[stray cypress]

👋 Welcome to your Help Thread!

Hey @shrewd bay, Thanks for sharing your math problem with us. While you wait for a Helper to help you, we want to share some vital information with you.

●

Please take a moment to read the helpee guidelines. This will make sure that your post follows the helpee rules of our community.

●

Please don't ping <@&775784618955505685>, <@&1283689826742440016>, <@&819616364188139550>, or <@&624327278137966593> for help because their job is to take care of the server's administrative tasks, not to answer queries directly. However, if you have a problem with how a Helper is acting, you can ping a Helper Moderator.

●

It's always very useful if you can show us the work you've done so far. This makes it easier for our Helpers to find mistakes and help you get to the right answer.

Once you've got your answer and you're all set, you can close this thread by typing +close, and then you can click the Thank buttons to show your Helper some appreciation. · May 29, 2026, 10:00 AM

[sly prairie]

Are you familiar with winding number of a curve?

The general residue theorem just multiplies the winding number with the residues

[shrewd bay]

Interesting, we have not covered that one in class so I'm afraid I can not use that

[sly prairie]

Okay but you asked if theres a shorter way

Without splitting thr contour

There is

Look it up, or wait for it to be covered in the class

[shrewd bay]

I meant more so; if there is a shorter way to see that the residues of 1 and -1 will cancel each other out. This was the final lecture, we will not cover anymore stuff.

[sly prairie]

Uhhh idts for a general function you need to calculate the residues

Theres no going around that

The winding numbers being 1 and 1 can be seen via drawing a line and from inside to out and counting the number of direction changed

[shrewd bay]

+close

[stray cypress]

Please thank your Helpers before closing!

Please thank the helpers who assisted you by clicking the buttons below. You can thank each helper only once. Once you're done, click "Close Post" to close this thread.

[stray cypress]

Thank you for your feedback! [redacted] has been awarded 1 . They now have 11 . They have 3 daily left for today.

[stray cypress]

@shrewd bay

<:HelpIcon:1304095958283321385>| Help Reminder

Hello .heavenorhell, this is a friendly reminder that your help request has been inactive for more than 24 hours. If you no longer need assistance, please consider closing the thread using the +close command. This thread will be automatically closed in 3 days if it remains inactive.