inverse fourier transform help

[a_ronagh]

unit step + inverse fourier

Hi everybody
I want to take an inverse fourier transform of a very big function.
I mean the integrand is very long and it covers 4 or 5 lines to write it only.
So I have to use numerical method to find it ,but there is another problem that
the function has a singularity at ω=0.I am really confused and I really need it for my project as well.I could not do it by MATLAB.
Please help me if you know a book or a free software explaning the methods of
integrating of complex functions.

thank you very much for your attention.

 

[DrDolittle]

inverse fourier of unit step

I got stuck with a similar problem. In my case, it was inverse laplace transform running for many lines for which neither matlab nor mathematica could help. In the first instance, check the individual elements making up the integrand is tranformable,atleast you can proceed with some kind of conviction.

Regards
drdolittle

 

[a_ronagh]

Thank you
After some large amount of manipulation, I could split my integrand into small inseparable parts which looks more like each other.I show one of these parts in the file that I have sent below.
Do you know how can I choose the path of integration in complex ky plane.
notice that there are two branch points and two poles.
branch points lie on the real axis but poles lie on the imaginary axis.

regards.

 

[steve10]

1. There is no branching point, but there are poles, especially the ones that make Q1 = 0. The poles are hard to get;
2. I don't see any term that can counter with ky^4 in the denominator. Therefore, your integral diverges no matter what you do.

 

[a_ronagh]

Thank you steve10.First I should correct an error in the doc file I have attached.
In definition of variable there is a 2 in the power.So we do have branch points. Another point is that how can you be so sure that this integral diverges?
For instance inverse fourier transform of 1/ky^4 is (y^3 /6)u(t).

Apart from the above problem,I have another question:
As I studied in IEEE papers,there are three main techniques in treating singularities that arises in sommerfeld integrals in full wave analysis:
contour-deformation approach,the folding technique and pole extraction method.
I could not find any discription about two latter ones.
Do you know some references about it?

 

[steve10]

I am sorry that I still don't see how you can spare a factor like ky^3 from the numerator, which is necessary to make the integral not diverge. Or perhaps, the integration is carried out from a-i*infinity to a+i*infinity?

I like the way using Green's functions in solving problems, but I suppose you are aware of how difficult to get them. It looks to me almost like you get everything when you get the Green's function, while you get nothing if you don't get it. Oh, well, good luck, then.

 

[steve10]

a_ronagh, you said " .... For instance inverse fourier transform of 1/ky^4 is (y^3 /6)u(t). "

You mean Fourier Inverse Transform (1/ky^4) = (y^3 /6)u(t)? Do you have a derivation? By the way, what is u(t)?

 

[a_ronagh]

Hi steve10
u(t) is the unit step function which has the value 1 for t>0 and zero for t<0.
I will send its derivation in complex ky plane for you.
I am sorry for my delay.

 

[steve10]

Hi steve10
u(t) is the unit step function which has the value 1 for t>0 and zero for t<0.
I will send its derivation in complex ky plane for you.
I am sorry for my delay.

That's ok, buddy, but I don't think you can really get one. Thanks for the clarification, though.

 

[hamidr_karami]

hi
this example for you

syms t u w x

ifourier(w*exp(-3*w)*sym('Heaviside(w)')) returns 1/2/pi/(3-i*t)^2


ifourier(1/(1 + w^2),u) returns


1/2*exp(-u)*Heaviside(u)+1/2*exp(u)*Heaviside(-u)


ifourier(v/(1 + w^2),v,u) returns i/(1+w^2)*Dirac(1,-u)


ifourier(sym('fourier(f(x),x,w)'),w,x) returns f(x)

bye