Expressing f(z) in terms of z

1. Jul 13, 2014

MissP.25_5

Hello! I am stuck at the final step. How do I get x+iy from the equation? Help!

I am so sorry for posting this question in a picture instead of writing it out, because I don't know how to write equations on here.

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2. Jul 13, 2014

Staff: Mentor

I see a small mistake: on the 4th line, shouldn't you be taking out the common factor 1/4?

Check your exponential equivalents for sine, cosine, etc., on line 3. There's your mistake.

Last edited: Jul 13, 2014
3. Jul 13, 2014

MissP.25_5

why 1/4 ?

4. Jul 13, 2014

Mentallic

You substituted incorrectly.

$$\cos{z}=\frac{e^{iz}+e^{-iz}}{2}$$

$$\sin{z}=\frac{e^{iz}-e^{-iz}}{2i}$$

And you should factor out a value on the 4th line because you want your expression to be of the form

$$k\left(\frac{e^{iz}-e^{-iz}}{2i}\right)$$
(or some similar form in the brackets)
where k is some complex number, and z is a complex number. Your expression would then simplify to $k\sin{z}$

5. Jul 13, 2014

MissP.25_5

But z=x+iy. How do I change i(x^2-y^2) into z? I can't figure out how to rearrange it.

6. Jul 13, 2014

Mentallic

Well of course you should recognize that

$$z^2=(x+iy)^2=x^2-y^2+i 2xy$$

and what you have is VERY similar to this form. Since you made a mistake early on that I pointed out, your final result isn't going to work with the method I'm hinting at here, but when you fix that up then it'll fall into place.

$$i(x^2-y^2)-2xy$$

Should be quite easily converted into a function of z by observing the z2 result.

7. Jul 14, 2014

MissP.25_5

Ok, so I have corrected my mistakes but I got stuck here. How do I continue?

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8. Jul 14, 2014

Mentallic

Now expand like you did earlier and simplify where possible.

9. Jul 14, 2014

MissP.25_5

I did that but still couldn't get the answer. Hold on, I will show you my work later.

10. Jul 14, 2014

Mentallic

Did you end up with

$$-i\left(\frac{e^{w}-e^{-w}}{2}\right)$$

where

$$w=i(x^2-y^2)-2xy$$

??

11. Jul 14, 2014

MissP.25_5

I don't know what to do next. How do I simplify this further?

Last edited: Jul 14, 2014
12. Jul 14, 2014

MissP.25_5

Sorry, did a mistake earlier. Here's what I got.

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13. Jul 14, 2014

Mentallic

Sorry, I only just glossed over your previous upload and didn't spot the error.

$$\sinh{z}=\frac{e^z-e^{-z}}{2}$$

while you had an i in the denominator.

14. Jul 14, 2014

MissP.25_5

Oh yes, that was my mistake. But then what happens to the i outside of the bracket?

15. Jul 14, 2014

Mentallic

The first 4 terms have a constant factor of

$$\frac{1}{4i}$$

while the right 4 are

$$\frac{i}{4}$$

How do these two numbers relate to each other?

16. Jul 14, 2014

MissP.25_5

They're minus to each other. Sorry, I don't know how to say that mathematically.

17. Jul 14, 2014

Mentallic

Right!

$$\frac{1}{4i}=\frac{i}{4(-1)}=\frac{-i}{4}$$

and so now you can continue to simplify the expression!

18. Jul 14, 2014

MissP.25_5

Simplified it, then what? I am still stuck here.

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19. Jul 14, 2014

Mentallic

Check post #10.

20. Jul 14, 2014

MissP.25_5

Uhmm, how do I change w into z? Can I just simpy do this?

exp(i*(ia-b)) ?? Is it possible to multiply i just like that?