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Integration result in terms of specific function?

Posted 3 years ago

I have a integral after convolution where there is a term of $Sin[\pi x]/x$ . I want the final integral in terms of Sinc[x] function. Is there a way to manipulate the expression so that possible Sin[pi x]/x gets replaced by Sinc.

Edit1- Suppose like I get the answer of the integral $$\exp(i2\pi ft)\frac{\sin(\pi k x)}{x} \;$$ I just want to convert to get the final answer in Sinc[x]. So the final answer that should come is $k\pi\exp(i2\pi ft) \text{Sinc}(\pi k x) \;$.

Also, apart from above is there something in Mathematica like rewrite (https://in.mathworks.com/help/symbolic/rewrite.html) command in Matlab.

Both like the examples are different. I was just trying to give examples of what I need. I just need whenever there is something like $\sin(k\; x)/x$ where k is a constant then it should get replaced by k*Sinc[x].

POSTED BY: Sahil Goyal
8 Replies

Maybe like so:

expr[Sin[Pi x]/x] /. Sin[a_] :> a Sinc[a]
POSTED BY: Henrik Schachner
Posted 3 years ago

I have edited the question. Can you please take a look?

POSTED BY: Sahil Goyal
Posted 3 years ago

It would really help people that are trying to help you if you post the WL code you have tried, not LaTeX.

ClearAll[f, t, k, x]
exp = Exp[2 Pi I f t] Sin[Pi k x]/x;

exp /. Sin[e_]/x -> Sinc[e]
(* E^(2 I f π t) Sinc[k π x] *)

Why does your desired output in LaTeX have a k π factor? That does not match the description of the desired output.

POSTED BY: Rohit Namjoshi
Posted 3 years ago

@Rohit Namjoshi I was just trying to give example. I was trying to say if I have anything like sin(k x)/x where k is a constant it should get replaced by a k Sinc[x].

POSTED BY: Sahil Goyal
Posted 3 years ago
exp /. Sin[k_ x]/x -> k Sinc[x]
POSTED BY: Rohit Namjoshi
Posted 3 years ago

Thanks. I didn't know that you can do something like Sin[k_ x]. I always used to think like thisSin[k_].

POSTED BY: Sahil Goyal

Welcome to Wolfram Community!
Please make sure you know the rules: https://wolfr.am/READ-1ST
Please describe your subject extensively providing minimized code.

POSTED BY: Moderation Team
Posted 3 years ago

I have edited the question. Can you take a look now if you know the answer.

POSTED BY: Sahil Goyal
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