Maximum of an integral that depends on a parameter

Asked Mar 15 '16 at 17:51

Active Mar 16 '16 at 03:31

Viewed 37 times

I have defined a function f(x,t) :

f[x_, t_] := 
NIntegrate[
  1/Sqrt[k^2 + 100] Exp[-I Sqrt[k^2 + 100] Exp[x] Cosh[t] + 
     I k Exp[x] Cosh[t] - I Log[(Sqrt[k^2 + 100] + k)/(Sqrt[k^2 + 100] - k)]],
 {k, -10, 10}]

Plotting it was not a problem (using Abs), but I was trying to get a better picture of it by looking for its maximum value, I couldn't obtain a result defining this function (it seems that to do the integral Mathematica requires a value for x and t):

g[t_] := MaxValue[Abs[f[x, t]], x]

Mathematica gave me results when I used:

FindMaximum[Abs[f[x, 4]], {x, -8}]

for particular values of t, but with errors. Is there a proper or better way to find the maximum of Abs[f] with respect to x for each value of t ? I thought of making a list (discretizing x and t), but maybe there are existing solutions inside Mathematica to deal with this sort of problems.

edited Mar 16 '16 at 03:31

J. M.'s missing motivation

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asked Mar 15 '16 at 17:51

faero

1

Define f using f[x_?NumericQ, t_?NumericQ]. – march Mar 15 '16 at 17:56
@march : What is the reason to put ?NumericQ after the arguments? – faero Mar 15 '16 at 18:09
It forces Mathematica to evaluate the function only when the arguments to the function are numbers. – march Mar 15 '16 at 18:17
@march But isn't there a way to make something like g(t) = MaxValue[f(x,t),x] ? – faero Mar 15 '16 at 18:26
2

g[t_?NumericQ] := First@FindMaximum[Abs[f[x, t]], {x, -8}]. – march Mar 15 '16 at 18:50

Maximum of an integral that depends on a parameter

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