Re: Problem with change of variables in an integral

• To: mathgroup at smc.vnet.net
• Subject: [mg131626] Re: Problem with change of variables in an integral
• From: Youngjoo Chung <ychung12 at gmail.com>
• Date: Fri, 13 Sep 2013 00:35:35 -0400 (EDT)
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• References: <l0mio3\$ltf\$1@smc.vnet.net>

```Dear Alexei,

I believe the references mentioned in Alex's reply below explains the evaluation procedure well.

f[z_] := 1/(1 + z^3);

Integrate[...] is evaluated to (2 \[Pi])/(3 Sqrt[3]) before being passed to Map. This explains the difference that you saw.

If f[z] is not known, Integrate cannot evaluate and is passed to Map as is as shown in your message. However, the following code

Clear[f, \[Phi]];
Map[ReplaceAll[#, {z -> r*Exp[I \[Phi]], \[Phi] -> 2 \[Pi]/3}] &,
Integrate[f[z], {z, 0, \[Infinity]}]]
% /. f[z_] -> z

issues an error message

Integrate::ilim: Invalid integration variable or limit(s) in {E^(I \[Phi]) r,0,\[Infinity]}. >>

and even though f[z_] = z is integrable, integration will not be done since E^(I \[Phi]) is not a valid integration variable.

Sincerely,
Youngjoo

```

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