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derivative of a well-behaved function

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  • Subject: [mg99466] derivative of a well-behaved function
  • From: Ricardo Samad <resamad at>
  • Date: Wed, 6 May 2009 05:24:52 -0400 (EDT)

Dear all,

the following ZS function is an analytical description of the transmittance
of a laser beam through an Iris after propagating inside a nonlinear sample
(Z-Scan curve):

\[Gamma][z_, z0_] := 1/2 (I/z0 (z + (z^2 + z0^2)/(DD - z)) + 1);

ZS[z_, z0_, \[Phi]_] :=
  Abs[\[Gamma][z, z0] Gamma[\[Gamma][z, z0], 0,
     I \[Phi]/(1 + (z/z0)^2)] /(I \[Phi]/(1 + (z/z0)^2))^\[Gamma][z,

Altough the function has imaginary arguments and is defined in terms of the
incomplete Gamma function, it is well-behaved and Mathematica calculates an=
plots it without problems:

DD = 300;
Plot[ZS[z, 1, 0.5], {z, -4, 4}]

The problem is that when I calculate its derivative in z, the result is
given in terms of infinite quantities and DirectInfinity functions, and it
is not possible to get numerical values of it neither plot its graph:

dZS = D[ZS[z, 1, 0.5], z];
dZS /. z -> 1
Plot[dZS[z, 1, 0.5], {z, -4, 4}]

Since the ZS function is well-behaved and has no discontinuities, its
derivative should be easily evaluated to numerical values and also plotted.
Does anybody has any idea on how to obtain the values? (I could easily writ=
a function to numerically calculate the derivative, but that=B4s not really
what I want).

Thank you,


Ricardo Elgul Samad

tel: (+55 11) 3133-9372
fax: (+55 11) 3133-9374

Centro de Lasers e Aplica=E7=F5es
AV. Prof. Lineu Prestes 2242
Cidade Universit=E1ria
S=E3o Paulo - SP

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