(omfg study hall)

tl;dr: Does sin(x) have a complex conjugate?

So, it's been awhile since I've worked with complex numbers. I'm working a problem where I need to normalize a function. Part of this is finding the complex conjugate of sin(x), where x is a real number. Finding the complex conjugate of a function, according to this professor (chemistry), is to switch i to -i.

So, does sin(x) have a complex conjugate? All of my google results involve situations where x is a complex number. Which is to say I've been finding , which if I have to find the complex conjugate of that, seems pretty damn annoying.

8/31/2009 10:48:53 PM

i've been finding fizzicksforums too

8/31/2009 10:49:43 PM

We don't do math problems that don't exist.

8/31/2009 10:50:37 PM

is this thread about pretending we can't do math, because i'm game

8/31/2009 10:52:46 PM

imagine that...

8/31/2009 10:52:46 PM

I feel like it would be [-i cos(x)].

8/31/2009 10:53:58 PM

probably need euler's equation or some shit like that

8/31/2009 10:54:57 PM

Yes, if x is always real, then the complex conjugate of sin(x) is sin(x).

To see that from your complex equation, write z := x = a + b*i. Then x is real, so b = 0.

Then

sin(x) = 1/2i * [e^(ia) - e^(-ia)]

so the complex conjugate is

1/2(-i) * [e^(-ia) - e^(ia)] = -1/2i * [e^(-ia) - e^(ia)] = 1/2i * [e^(ia) - e^(-ia)] = sin(x)

[Edited on August 31, 2009 at 10:59 PM. Reason : proofage]

8/31/2009 10:56:26 PM

i'm going to go with the old math teacher goto excuse for things you aren't supposed to learn about yet and say that you can't do that and it doesn't exist. sin(x) isn't defined for imaginary numbers

8/31/2009 10:56:54 PM

8/31/2009 10:58:24 PM

Nice.

8/31/2009 10:59:14 PM

haha i knew there was some euler's eqn shit in there

8/31/2009 11:01:45 PM

Quote :

"Yes, if x is always real, then the complex conjugate of sin(x) is sin(x). [math]"

Cool, thanks for the help! I wrote that out, but didn't recognize the equivalency of the conjugate before.

8/31/2009 11:09:13 PM

imaginary math solution below:

8/31/2009 11:12:20 PM

paging InsaneMan

8/31/2009 11:12:55 PM

No problem.

[Edited on August 31, 2009 at 11:14 PM. Reason : InsaneMan would write an AI routine in Java to numerically derive it using genetic algorithms.]

8/31/2009 11:14:30 PM