anonymous 5 years ago Find the holomorphic function f(x+iy) such that Re f(x+iy)= cosh(3y)sin(3x) and f(0)=0

1. anonymous

ugh i havent taken calc yet so i cant do this

2. anonymous

cauchy reimann yes?

3. anonymous

yes satellite

4. anonymous

so the derivative of this wrt x must be the derivative of the imaginary part wrt y right? the derivative is $3\cos(3x)\cosh(3y)$ integrate wrt y and get $\cos(3x)\sinh(3y)$ i think.

5. anonymous

it's been a while. am i on the right track?

6. anonymous

you also need the partial wrt y is minus the partial wrt x of the imaginary part. if we are lucky it already is oh and you also need that f(0)=0 i forgot the +C when i integrated

7. anonymous

how to you like that? i miracle! it works

8. anonymous

so i guess unless i totally screwed this up the answer is $f(x+iy)=\cosh(3y)\sin(3x)+\cos(3x)\sinh(3y)$

9. anonymous

think we lost raheen maybe he will be back

10. anonymous

rather $f(x+iy)=\cosh(3y)\sin(3x)+i\cos(3x)\sinh(3y)$

11. anonymous

i forgot the i part

12. anonymous

@raheen look ok?

13. anonymous

it does not come with a money guarantee because it has been several years but i think it looks good. you can check cauchy reimann equations for this and see if they work

14. anonymous

satellite you are about to reach , great work I think you did how about using the 2 conditions of C-R then you need to integrate and don forget to find the constant

15. anonymous

well i integrated wrt y and got the answer. then i checked that the second equation and it worked. and by inspection you can see that $f(0)=0$

16. anonymous

i mean if the C-R equations are going to work, then after i take the derivative wrt x and integrate wrt y , then the second condition $\frac{\delta u}{\delta y}=-\frac{\delta v}{\delta x}$ had better work or else we are screwed. both conditions must hold. but in any case i checked and they do

17. anonymous

that's so great satellite, thank you.

18. anonymous

yw

19. Zarkon

the only thing I would add is that when you integrated with respect to y you don't get C you get some function of x

20. anonymous

Zarkon, thank you, you are right it's C(x)

21. Zarkon

It doesn't look like it will change the final answer though