Showing Alternative Methods for Solving Problems

  • Thread starter Shevchenko
  • Start date
In summary: I don't know about complex contour integration can u show me the solutionNo, I'll give you hints but I can't give the complete solution.
  • #1
Shevchenko
4
0
show that (don't use Fourier-Transformation method):
 

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  • #2
Do you know complex contour integration?
 
  • #3
i don't know about complex contour integration
can u show me the solution
 
  • #4
No, I'll give you hints but I can't give the complete solution. What method are you supposed to use?
 
  • #5
Shevchenko said:
i don't know about complex contour integration

What do you know then?

Shevchenko said:
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.
 
  • #6
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
 
  • #7
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.
 
  • #8
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?
 

1. What is the importance of showing alternative methods for solving problems?

Showing alternative methods for solving problems is important because it allows for a more comprehensive understanding of the problem and potential solutions. It also encourages critical thinking and creativity, as different methods may have unique advantages and drawbacks.

2. How do you determine which alternative method is the most effective?

The most effective alternative method for solving a problem depends on a variety of factors, such as the specific problem, available resources, and personal preferences. It is important to carefully consider and evaluate each method before deciding on the most suitable one.

3. Can alternative methods for solving problems be applied in all scientific fields?

Yes, alternative methods for solving problems can be applied in all scientific fields. While the specific methods may vary, the concept of exploring different approaches to problem-solving is applicable in all areas of science.

4. How do you present alternative methods in a scientific paper or presentation?

In a scientific paper or presentation, alternative methods for solving problems should be clearly and concisely described, along with their potential advantages and limitations. Visual aids, such as diagrams or graphs, can also be useful in illustrating the different methods.

5. Can alternative methods for solving problems lead to unexpected discoveries?

Yes, alternative methods for solving problems can lead to unexpected discoveries. By exploring different approaches, scientists may stumble upon new and innovative solutions that they may not have otherwise considered. This is why it is important to always keep an open mind and consider alternative methods.

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