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I asked Wolfram Alpha to give me a solution to an integral function https://www.wolframalpha.com/input/?i=(integral+exp(-mx)+dx+between+x-a+and+x%2Ba+)%2F(2+a+exp(-mx)) and it gave me an expression which is equivalent to $z=\frac{\sinh(xy)}{xy}$.

It also produced a contour plot of the value of $z$ for the domain $[-0.002<x<0.002,-0.002<y<0.002]$. The surface z is not smooth but has a complex (fractal?) appearance.

When, using the formula $z = \sinh(x)/x$, I calculated values in Excel I found a similar non-smooth behavior for small values of $x:$ ($0<x<0.00002$).

So my question is: Does $z=\sinh(x)/x$ have a fractal behavior for small values of $x$, or is the irregularity (displayed in Wolfram Alpha and Excel) due to rounding errors in the computational engines?

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    $\begingroup$ No, in fact $\frac{\sinh x}{x}$ has a removable singularity at $x = 0$, and removing the singular gives a function which is analytic. Expanding using the usual power series for $\sinh x$ gives $\frac{\sinh x}{x} = 1 + O(x^2)$. $\endgroup$ – Travis Willse Feb 20 at 1:20
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    $\begingroup$ @Travis Ah I forgot to look for a power series expansion of sinh(x). Thanks. $\endgroup$ – steveOw Feb 20 at 1:53

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