MHB Fourier and inverse fourier transform

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The discussion centers on the Fourier and inverse Fourier transforms, with a request for resources that provide both theoretical explanations and solved examples. The user expresses confusion about when to apply the definition involving integration versus using shortcut formulas from reference sheets. There is a recognition that deriving results from fundamental definitions is beneficial, but it's acceptable to use tabulated values when appropriate. The conversation emphasizes the importance of understanding the underlying principles while also being pragmatic in problem-solving approaches. Overall, balancing thorough comprehension with practical application is key in mastering Fourier transforms.
goohu
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Hi, I've been looking all over the net for good examples but I've only found some intro but no examples being solved.

If you know of good resources (both theories and problems) please let me know!

View attachment 9435

a) Calculate Fourier and inverse Fourier transform of f(t).
b) Calculate the limit.

My question is :
it looks like sometimes you use the definition of the transform with the integration of f(t) * e^-iwt ,
other times you use shortcuts from formula sheets. Any tips of when to use which?

I'm pretty clueless about this one right now , is it okay to just ask for a solution with explanation?
 

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I use the way, which I like more. Sometimes I'm "lazy" and take tabular values, sometimes I derive the results from definitions and theorems. So, do what suits you, your needs and situation! It's good to know, how the most essential results are derived, but it's not necessarily required to derive everything again in the case of new problem.
 
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