How evaluate this f(x)=x^2+(1)/(1+2x^4) with fourier tranform

ormaybesaladqh 2022-10-23 Answered
How evaluate this f ( x ) = x 2 + 1 1 + 2 x 4 with fourier tranform
You can still ask an expert for help

Want to know more about Laplace transform?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

ehedem26
Answered 2022-10-24 Author has 13 answers
The Fourier transform of f(x) doesn't exist in the usual sense, but since f can be viewed as a tempered distribution, we can interpret the Fourier transform in that setting. (I'm using the normalization f ^ ( ω ) = e i ω t f ( t ) d t. If you're using something else, the answer is a little different.)
First of all, Fourier transform of 1 is 2 π δ ( ω ). Hence
t F 2 π i δ ( ω ) t 2 F 2 π i 2 δ ( ω ) = 2 π δ ( ω ) .
The second term is less problematic, and exists in the usual sense. It is a standard exercise in residue calculus to compute the Fourier transform of 1 1 + 2 x 4 . The result (and especially all the intermediate steps) are very messy though. I get:
f ^ ( ω ) = 2 π δ ( ω ) + { π q e q ω ( cos q ω sin q ω ) , ω < 0 π q e q ω ( sin q ω cos q ω ) , ω 0
where q = 2 1 / 4 / 2
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

New questions