# Is the function $\sinh(x)/x$ fractal at small values of $x,y$ or am I seeing rounding errors in computation?

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. 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).

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?

• 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 $\frac{\sinh x}{x} = 1 + O(x^2)$. – Travis Willse Feb 20 at 1:20
• @Travis Ah I forgot to look for a power series expansion of sinh(x). Thanks. – steveOw Feb 20 at 1:53