Question

Complex analysis!
Please help!

Answer 1

\[\int\limits_{0}^{\infty} \frac{ xsinx dx }{ 9x^2+4 }\]

Answer 2

use contour integration

Answer 3

@ganeshie8
@dan815
@iambatman

Answer 4

can u explain contour integration? do u want me to take this to complex domain?

Answer 5

yes x---> z

Answer 6

so we use both real and imaginary parts to make a contour and integrate over it

Answer 7

err rather we transform this real function into a complex one

Answer 8

@Kainui @satellite73 @hartnn @ganeshie8

Answer 9

well, I can explain what you normally do if that would help. I'm just stuck on this problem because its different from the rest because of the x in the numerator

Answer 10

ok ya i am interested to know

Answer 11

yeah it is not the x in the numerator, it is the sine
how do you handle that?

Answer 12

I know cosx-->e^iz
not sure about the sin though

Answer 13

reason is that you break up your complex integral into two parts: real and imaginary
real on x, imaginary on the semi circle

Answer 14

god do i love google!!

Answer 15

look at this nice example 4, where it is
\[\int_0^{\infty}\frac{x\sin(x)}{x^2+1}dx\] couldn’t ask for a better template

Answer 16

lol awesome

Answer 17

oh hey I have the hardcopy of that text book :D

Answer 18

why can they change the limits of integration if sin is not even

Answer 19

it all (almost) comes back to me
you just integrate the whole damned thing, then you equate the real and imaginary parts to get the integral with sine and cosine

Answer 20

sine is not even, but the integrand is

Answer 21

how?

Answer 22

cause it is

Answer 23

you got the x in front of the sine

Answer 24

ooh right

Answer 25

you can follow along this example exactly
at some point you are going to replace \(3i\) by \(\frac{2}{3}i\) and instead of getting
\[\frac{\pi}{2e^3}\] you will get
\[\frac{\pi}{18e^{\frac{2}{3}}}\]

Answer 26

I get that now. So can you explain why sin x doesn't--->sin z
instead sin x ---> e^iz

Answer 27

also what did you type into google to find these examples

Answer 28

my secret

Answer 29

noooooooo

Answer 30

:'(

Answer 31

if they had had this when i was in school, i have had like 3 PhD's by now

Answer 32

residue to compute integral xsin(x)/(9x^2+4)

Answer 33

the gimmick is not to replace sine by anything, just change the whole thing to \[\int\frac{x}{9x^2+4}e^{ix}dx\]
compute that sucker then split in to real and imaginary parts since
\[e^{ix}=\cos(x)+i\sin(x)\]

Answer 34

with e^ix= cosx+isinx the sin is imaginary right?
sin in this problem is real

Answer 35

you compute the whole thing
then you equate the real part to the real part, and the imaginary part to the imaginary part
like if the integral turned out to be \(2+3i\) then the real part is the part with cosine, that would be 2, and the imaginary part is the part with sine, that would be 3

Answer 36

oh I see

Answer 37

so regardless if the numerator is sin or cos you turn the thing into e^iz

Answer 38

sine isn't imaginary, i is
in \[e^{ix}=\cos(x)+i\sin(x)\] sine and cosine are real, only the i is imaginary and sine is the imaginary part

Answer 39

oh okay that makes sense

Answer 40

not \(e^{iz}\) just \(e^{ix}\)

Answer 41

oh my book turns it into e^iz

Answer 42

hmm ok

Answer 43

it splits up the real part of z and im part of y

Answer 44

z*

Answer 45

thanks for your help satellite