Sine integral and hyperbolic tangent.

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Why does $operatornameSi(x)$ resemble $tanh(x)(1+fracsin xx)$? How are both functions related? What is the simplest function that produces such a shape? What are the uses of such a function, and where does it arise naturally? Can such a function be used as an activation function in neural nets?




special-functions




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    And even more similar function is $left(fracpi2+fracsin xxright)tanh x $
    – gammatester
    2 days ago







  • 1




    When you write textSi instead of operatornameSi, then you don't get proper spacing in things like $5operatornameSi x$ and $5operatornameSi(x),$ and instead you see $5textSi x$ and $5textSi(x).$ I include both examples in order to show the context-dependent nature of the spacing, i.e. in the two that are done correctly, one of them has more space to the right of $operatornameSi$ than the other one.
    – Michael Hardy
    2 days ago











  • $x mapsto left(fracpi2 + frac1-cos(x)xright) tanh(x)$ almost has the same maximum and minimum points as $operatornameSi$ and the correct limit as $x to infty$. $x mapsto left(fracfracpi2(1+x)fracpi2+x + frac1-cos(x)xright) tanh(x)$ is even better.
    – ComplexYetTrivial
    yesterday











  • The Fourier transform of the Sinc function, $fracsin xx$, is a simple rectangular function (the ideal low pass filter), with the Fourier Transform of its integral, $operatornameSi(x)$, being related to this. What might be interesting is to differentiate $tanh(x)(1+fracsin xx)$ and calculate its Fourier transform to clarify the difference in the frequency domain. Using Mathematica I have managed to calculate the Fourier transform of $tanh(x)(1+fracsin xx)$, but not its derivative unfortunately.
    – James Arathoon
    yesterday















up vote
2
down vote

favorite
1












Why does $operatornameSi(x)$ resemble $tanh(x)(1+fracsin xx)$? How are both functions related? What is the simplest function that produces such a shape? What are the uses of such a function, and where does it arise naturally? Can such a function be used as an activation function in neural nets?




special-functions




share|cite|improve this question

















  • 2




    And even more similar function is $left(fracpi2+fracsin xxright)tanh x $
    – gammatester
    2 days ago







  • 1




    When you write textSi instead of operatornameSi, then you don't get proper spacing in things like $5operatornameSi x$ and $5operatornameSi(x),$ and instead you see $5textSi x$ and $5textSi(x).$ I include both examples in order to show the context-dependent nature of the spacing, i.e. in the two that are done correctly, one of them has more space to the right of $operatornameSi$ than the other one.
    – Michael Hardy
    2 days ago











  • $x mapsto left(fracpi2 + frac1-cos(x)xright) tanh(x)$ almost has the same maximum and minimum points as $operatornameSi$ and the correct limit as $x to infty$. $x mapsto left(fracfracpi2(1+x)fracpi2+x + frac1-cos(x)xright) tanh(x)$ is even better.
    – ComplexYetTrivial
    yesterday











  • The Fourier transform of the Sinc function, $fracsin xx$, is a simple rectangular function (the ideal low pass filter), with the Fourier Transform of its integral, $operatornameSi(x)$, being related to this. What might be interesting is to differentiate $tanh(x)(1+fracsin xx)$ and calculate its Fourier transform to clarify the difference in the frequency domain. Using Mathematica I have managed to calculate the Fourier transform of $tanh(x)(1+fracsin xx)$, but not its derivative unfortunately.
    – James Arathoon
    yesterday













up vote
2
down vote

favorite
1









up vote
2
down vote

favorite
1






1





Why does $operatornameSi(x)$ resemble $tanh(x)(1+fracsin xx)$? How are both functions related? What is the simplest function that produces such a shape? What are the uses of such a function, and where does it arise naturally? Can such a function be used as an activation function in neural nets?




special-functions




share|cite|improve this question













Why does $operatornameSi(x)$ resemble $tanh(x)(1+fracsin xx)$? How are both functions related? What is the simplest function that produces such a shape? What are the uses of such a function, and where does it arise naturally? Can such a function be used as an activation function in neural nets?





special-functions





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited 2 days ago









Michael Hardy

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user575696

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  • 2




    And even more similar function is $left(fracpi2+fracsin xxright)tanh x $
    – gammatester
    2 days ago







  • 1




    When you write textSi instead of operatornameSi, then you don't get proper spacing in things like $5operatornameSi x$ and $5operatornameSi(x),$ and instead you see $5textSi x$ and $5textSi(x).$ I include both examples in order to show the context-dependent nature of the spacing, i.e. in the two that are done correctly, one of them has more space to the right of $operatornameSi$ than the other one.
    – Michael Hardy
    2 days ago











  • $x mapsto left(fracpi2 + frac1-cos(x)xright) tanh(x)$ almost has the same maximum and minimum points as $operatornameSi$ and the correct limit as $x to infty$. $x mapsto left(fracfracpi2(1+x)fracpi2+x + frac1-cos(x)xright) tanh(x)$ is even better.
    – ComplexYetTrivial
    yesterday











  • The Fourier transform of the Sinc function, $fracsin xx$, is a simple rectangular function (the ideal low pass filter), with the Fourier Transform of its integral, $operatornameSi(x)$, being related to this. What might be interesting is to differentiate $tanh(x)(1+fracsin xx)$ and calculate its Fourier transform to clarify the difference in the frequency domain. Using Mathematica I have managed to calculate the Fourier transform of $tanh(x)(1+fracsin xx)$, but not its derivative unfortunately.
    – James Arathoon
    yesterday













  • 2




    And even more similar function is $left(fracpi2+fracsin xxright)tanh x $
    – gammatester
    2 days ago







  • 1




    When you write textSi instead of operatornameSi, then you don't get proper spacing in things like $5operatornameSi x$ and $5operatornameSi(x),$ and instead you see $5textSi x$ and $5textSi(x).$ I include both examples in order to show the context-dependent nature of the spacing, i.e. in the two that are done correctly, one of them has more space to the right of $operatornameSi$ than the other one.
    – Michael Hardy
    2 days ago











  • $x mapsto left(fracpi2 + frac1-cos(x)xright) tanh(x)$ almost has the same maximum and minimum points as $operatornameSi$ and the correct limit as $x to infty$. $x mapsto left(fracfracpi2(1+x)fracpi2+x + frac1-cos(x)xright) tanh(x)$ is even better.
    – ComplexYetTrivial
    yesterday











  • The Fourier transform of the Sinc function, $fracsin xx$, is a simple rectangular function (the ideal low pass filter), with the Fourier Transform of its integral, $operatornameSi(x)$, being related to this. What might be interesting is to differentiate $tanh(x)(1+fracsin xx)$ and calculate its Fourier transform to clarify the difference in the frequency domain. Using Mathematica I have managed to calculate the Fourier transform of $tanh(x)(1+fracsin xx)$, but not its derivative unfortunately.
    – James Arathoon
    yesterday








2




2




And even more similar function is $left(fracpi2+fracsin xxright)tanh x $
– gammatester
2 days ago





And even more similar function is $left(fracpi2+fracsin xxright)tanh x $
– gammatester
2 days ago





1




1




When you write textSi instead of operatornameSi, then you don't get proper spacing in things like $5operatornameSi x$ and $5operatornameSi(x),$ and instead you see $5textSi x$ and $5textSi(x).$ I include both examples in order to show the context-dependent nature of the spacing, i.e. in the two that are done correctly, one of them has more space to the right of $operatornameSi$ than the other one.
– Michael Hardy
2 days ago





When you write textSi instead of operatornameSi, then you don't get proper spacing in things like $5operatornameSi x$ and $5operatornameSi(x),$ and instead you see $5textSi x$ and $5textSi(x).$ I include both examples in order to show the context-dependent nature of the spacing, i.e. in the two that are done correctly, one of them has more space to the right of $operatornameSi$ than the other one.
– Michael Hardy
2 days ago













$x mapsto left(fracpi2 + frac1-cos(x)xright) tanh(x)$ almost has the same maximum and minimum points as $operatornameSi$ and the correct limit as $x to infty$. $x mapsto left(fracfracpi2(1+x)fracpi2+x + frac1-cos(x)xright) tanh(x)$ is even better.
– ComplexYetTrivial
yesterday





$x mapsto left(fracpi2 + frac1-cos(x)xright) tanh(x)$ almost has the same maximum and minimum points as $operatornameSi$ and the correct limit as $x to infty$. $x mapsto left(fracfracpi2(1+x)fracpi2+x + frac1-cos(x)xright) tanh(x)$ is even better.
– ComplexYetTrivial
yesterday













The Fourier transform of the Sinc function, $fracsin xx$, is a simple rectangular function (the ideal low pass filter), with the Fourier Transform of its integral, $operatornameSi(x)$, being related to this. What might be interesting is to differentiate $tanh(x)(1+fracsin xx)$ and calculate its Fourier transform to clarify the difference in the frequency domain. Using Mathematica I have managed to calculate the Fourier transform of $tanh(x)(1+fracsin xx)$, but not its derivative unfortunately.
– James Arathoon
yesterday





The Fourier transform of the Sinc function, $fracsin xx$, is a simple rectangular function (the ideal low pass filter), with the Fourier Transform of its integral, $operatornameSi(x)$, being related to this. What might be interesting is to differentiate $tanh(x)(1+fracsin xx)$ and calculate its Fourier transform to clarify the difference in the frequency domain. Using Mathematica I have managed to calculate the Fourier transform of $tanh(x)(1+fracsin xx)$, but not its derivative unfortunately.
– James Arathoon
yesterday
















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