Re: derivative of a well-behaved function

• To: mathgroup at smc.vnet.net
• Subject: [mg99520] Re: derivative of a well-behaved function
• From: dh <dh at metrohm.com>
• Date: Thu, 7 May 2009 06:34:34 -0400 (EDT)
• References: <gtrl0i\$1mf\$1@smc.vnet.net>

```
Hi Ricardo,

I can not tell what is going wrong. But you could calcuölate an

approximate function and get the derivative of it. Herei is your example:

==================================

fun = FunctionInterpolation[ZS[z, 1, 0.5], {z, -4, 4}];

Plot[{fun[z], ZS[z, 1, 0.5]}, {z, -4, 4}]

Plot[fun'[z], {z, -4, 4}]

=================================

Daniel

> 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,

>      z0]]^2;

>

> Altough the function has imaginary arguments and is defined in terms of the

> incomplete Gamma function, it is well-behaved and Mathematica calculates an=

> d

> 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

> N[%]

> 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=

> e

> a function to numerically calculate the derivative, but that=B4s not really

> what I want).

>

> Thank you,

>

> Ricardo

>

>

> --

> ____________________________________

>

> tel: (+55 11) 3133-9372

> fax: (+55 11) 3133-9374

>

> Centro de Lasers e Aplica=E7=F5es

> IPEN/CNEN-SP

> AV. Prof. Lineu Prestes 2242

> 05508-000

> S=E3o Paulo - SP

> Brazil

> ____________________________________

>

>

```

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