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Why does $\operatorname{Si}(x)$ resemble $\tanh(x)(1+\frac{\sin x}{x})$? 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?

Michael Hardy
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user575696
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    And even more similar function is $\left(\frac{\pi}{2}+\frac{\sin x}{x}\right)\tanh x $ – gammatester Aug 04 '18 at 13:11
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    When you write \text{Si} instead of \operatorname{Si}, then you don't get proper spacing in things like $5\operatorname{Si} x$ and $5\operatorname{Si}(x),$ and instead you see $5\text{Si} x$ and $5\text{Si}(x).$ 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 $\operatorname{Si}$ than the other one. – Michael Hardy Aug 04 '18 at 14:22
  • $x \mapsto \left(\frac{\pi}{2} + \frac{1-\cos(x)}{x}\right) \tanh(x)$ almost has the same maximum and minimum points as $\operatorname{Si}$ and the correct limit as $x \to \infty$. $x \mapsto \left(\frac{\frac{\pi}{2}(1+x)}{\frac{\pi}{2}+x} + \frac{1-\cos(x)}{x}\right) \tanh(x)$ is even better. – ComplexYetTrivial Aug 05 '18 at 00:43
  • The Fourier transform of the Sinc function, $\frac{\sin x}{x}$, is a simple rectangular function (the ideal low pass filter), with the Fourier Transform of its integral, $\operatorname{Si}(x)$, being related to this. What might be interesting is to differentiate $\tanh(x)(1+\frac{\sin x}{x})$ and calculate its Fourier transform to clarify the difference in the frequency domain. Using Mathematica I have managed to calculate the Fourier transform of $\tanh(x)(1+\frac{\sin x}{x})$, but not its derivative unfortunately. – James Arathoon Aug 05 '18 at 15:33

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