# Function approaching incorrect limit in mathematica [closed]

I am interested in the complex function

$f(z)=\frac{1}{a}\log\left[2\sinh(a\sqrt{z})\right]$.

where $a > 0$ is a real parameter. Clearly for large $a$ this approaches $f(z) \to \sqrt{z}$.

But Mathematica suggests otherwise. On the real axis the function approaches the correct limit but in the complex plane the function is plagued by discontinuities as $a$ becomes large.

a = 0.9;
Plot[{Log[2 Sinh[a Sqrt[z]]]/a, Sqrt[z]}, {z, 0, 10}, PlotRange -> {0, 5}]
Plot3D[Im[Sqrt[x + I y]], {x, -100, 100}, {y, -100, 100}]
Plot3D[Im[Log[2 Sinh[a Sqrt[x + I y]]]/a], {x, -100, 100}, {y, -100, 100}]


Is there any known way to overcome this problem in Mathematica?

Have a look at the result of plotting Im[Log[2 Sinh[a Sqrt[z]]]/a] and Im[Log[2 Sinh[-a Sqrt[z]]]/a] together... –  Ｊ. Ｍ. Nov 17 '12 at 18:46
Even though $a$ is supposed to be real, you need to look at the behavior of $f(z)$ for complex values of $a$ to understand this limit. To study it, use $1/a$ instead of $a$ and look near the origin. It's a wonderful picture, well worth displaying: f[a_] := Log[2 Sinh[1/a]] a; With[{z = 1/4}, ContourPlot[Abs[f[x + I y]], {x, -z, z}, {y, -z, z}, PlotPoints -> 50]]. (Replace Abs by Arg to see the argument of $f$.) –  whuber Nov 18 '12 at 22:56