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Fourier transform of $mathrme^-x^2 + i x^4$

-2

$begingroup$

I would like to know if there is an analytical solution to the Fourier transform of
$$mathrme^-x^2 + i x^4,,$$
where $i$ is the imaginary unit.

More generally, is there an analytical solution to the Fourier transform of a function

$$mathrme^p_1(x) + i p_2(x),,$$

where $p_1(x)$ and $p_2(x)$ are polynomial functions of orders $n_1$ and $n_2$?

Thank you.

complex-analysis fourier-analysis linear-transformations

share|cite|improve this question

edited Mar 19 at 10:25

Peiffap

10312

asked Mar 19 at 10:05

Štěpán VenosŠtěpán Venos

1

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  $begingroup$
  Do you have any reason to think that it can be computed explicitly?
  $endgroup$
  – Kavi Rama Murthy
  Mar 19 at 10:07
  

  
  
  
  


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  WolframAlpha can't find a solution to $mathscrF_xleft[mathrme^-x^2 + mathrmix^4right](omega)$, so I don't think there is a closed form answer to your question.
  $endgroup$
  – Peiffap
  Mar 19 at 10:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Maybe you can complete square, square substitute and Fourier-transform a gaussian. The other idea would be to write $e^p_1(x)cdot e^ip_2(x)$ and use convolution theorem. If you can find Fourier transform of any exponential family function.
  $endgroup$
  – mathreadler
  Mar 19 at 10:58
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  $e^−x^2+i*x^2$ can be computed explicitly so I think there could be solution also for higher powers
  $endgroup$
  – Štěpán Venos
  Mar 19 at 11:26
  

  
  
  
  

add a comment | 

-2

$begingroup$

I would like to know if there is an analytical solution to the Fourier transform of
$$mathrme^-x^2 + i x^4,,$$
where $i$ is the imaginary unit.

More generally, is there an analytical solution to the Fourier transform of a function

$$mathrme^p_1(x) + i p_2(x),,$$

where $p_1(x)$ and $p_2(x)$ are polynomial functions of orders $n_1$ and $n_2$?

Thank you.

complex-analysis fourier-analysis linear-transformations

share|cite|improve this question

edited Mar 19 at 10:25

Peiffap

10312

asked Mar 19 at 10:05

Štěpán VenosŠtěpán Venos

1

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Do you have any reason to think that it can be computed explicitly?
  $endgroup$
  – Kavi Rama Murthy
  Mar 19 at 10:07
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  WolframAlpha can't find a solution to $mathscrF_xleft[mathrme^-x^2 + mathrmix^4right](omega)$, so I don't think there is a closed form answer to your question.
  $endgroup$
  – Peiffap
  Mar 19 at 10:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Maybe you can complete square, square substitute and Fourier-transform a gaussian. The other idea would be to write $e^p_1(x)cdot e^ip_2(x)$ and use convolution theorem. If you can find Fourier transform of any exponential family function.
  $endgroup$
  – mathreadler
  Mar 19 at 10:58
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  $e^−x^2+i*x^2$ can be computed explicitly so I think there could be solution also for higher powers
  $endgroup$
  – Štěpán Venos
  Mar 19 at 11:26
  

  
  
  
  

add a comment | 

$begingroup$

I would like to know if there is an analytical solution to the Fourier transform of
$$mathrme^-x^2 + i x^4,,$$
where $i$ is the imaginary unit.

More generally, is there an analytical solution to the Fourier transform of a function

$$mathrme^p_1(x) + i p_2(x),,$$

where $p_1(x)$ and $p_2(x)$ are polynomial functions of orders $n_1$ and $n_2$?

Thank you.

complex-analysis fourier-analysis linear-transformations

share|cite|improve this question

edited Mar 19 at 10:25

Peiffap

10312

asked Mar 19 at 10:05

Štěpán VenosŠtěpán Venos

1

$endgroup$

share|cite|improve this question

edited Mar 19 at 10:25

Peiffap

10312

asked Mar 19 at 10:05

Štěpán VenosŠtěpán Venos

1

share|cite|improve this question

edited Mar 19 at 10:25

Peiffap

10312

asked Mar 19 at 10:05

Štěpán VenosŠtěpán Venos

1

edited Mar 19 at 10:25

Peiffap

10312

edited Mar 19 at 10:25

Peiffap

10312

asked Mar 19 at 10:05

Štěpán VenosŠtěpán Venos

1

asked Mar 19 at 10:05

Štěpán VenosŠtěpán Venos

1