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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?

Thanks in advance.

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7  
Those are just the different branches of your multivalued complex function. – rm -rf♦ Nov 17 '12 at 18:38
1  
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
1  
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

closed as too localized by whuber, Silvia, Sjoerd C. de Vries, m_goldberg, Yves Klett Apr 30 at 6:34

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