Showing Alternative Methods for Solving Problems

[Shevchenko]

show that (don't use Fourier-Transformation method):

 

[StatusX]

Do you know complex contour integration?

 

[Shevchenko]

i don't know about complex contour integration
can u show me the solution

 

[StatusX]

No, I'll give you hints but I can't give the complete solution. What method are you supposed to use?

 

[shmoe]

i don't know about complex contour integration

What do you know then?

can u show me the solution

We don't just give out solutions here. You have to show some work. Show us what you've tried, where you are stuck and why. You'll get much more help if we are convinced you're putting effort into this.

 

[Shevchenko]

i used
cos ax=.(e^ajx + e^-ajx)/2 ,where lambda=a
then i put
jax=u at the first positive part
i got the following integration
a/2j int[(e^u)/(a^4-u^2)] respect to u
i stopped here : i can't evaluate this integral by parts
give me the name of method here only
thanks for helping

 

[StatusX]

The indefinite integral cannot be (finitely) expressed in terms of elementary functions. I can't really think of a way to do this besides contour integration. Fourier transforms might work, although I'm not exactly sure how, and I can't imagine why you wouldn't be allowed to use them if they did give the answer. Maybe someone else has some ideas.

 

[shmoe]

Your integrand is even, so you can change the limits of integration to (-infinity,infinity) and take half this value. Then replace cos(lambda*x) with exp(i*lambda*x), the imaginary part will be zero from symmetry. You can then write this in terms of a common Fourier transfom pair that could be looked up on a table or derived, see the "Exponential Function" on the table in http://mathworld.wolfram.com/FourierTransform.html.

One direction can be derived easily with some basic calculus(going from f(x) to F(k) in the table), but the other direction is the one you need unless you can invoke something to do with inverse Fourier transforms. I'm not sure how to derive this direction without using contour integration (or inverting Fourier transforms). Why must you avoid Fourier transform techniques?