Characteristic Function Integrand Evaluation

[CivilSigma]

Homework Statement

[/B]
I am trying to determien the characteristic function of the function:

$$ f(x)= ae^{-ax}$$

$$\therefore E(e^{itx}) =\int_0^\infty e^{itx}ae^{-ax} dx = a \cdot \frac{e}{it-a} |_0 ^ \infty $$

But I am not sure how to evaluate the integral.

Wolfram alpha suggests this, but I am not sure how to get there.

https://www.wolframalpha.com/input/?i=integral+from+0+to+infitiy+of+e^(itx)*ae^(-ax)dx

The Attempt at a Solution

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If you just plug in the limits you get (∞ - ...) which is indeterminate.

 

[Ray Vickson]

Homework Statement

[/B]
I am trying to determien the characteristic function of the function:

$$ f(x)= ae^{-ax}$$

$$\therefore E(e^{itx}) =\int_0^\infty e^{itx}ae^{-ax} dx = a \cdot \frac{e}{it-a} |_0 ^ \infty $$

But I am not sure how to evaluate the integral.

Wolfram alpha suggests this, but I am not sure how to get there.

https://www.wolframalpha.com/input/?i=integral+from+0+to+infitiy+of+e^(itx)*ae^(-ax)dx

The Attempt at a Solution

[/B]
If you just plug in the limits you get (∞ - ...) which is indeterminate.

You have already evaluated the integral; it is not at all indeterminate. If $F(t) = \int f(t) \, dt$ is the indefinite integral, then $\int_0^\infty f(t) \, dt = \lim_{t \to \infty} F(t) - F(0).$ Remember: you were supposed to have $a > 0,$ but you did not actually say that when you wrote the question.​

 

[Mark44]

I am trying to determien the characteristic function of the function:
$$ f(x)= ae^{-ax}$$

$$\therefore E(e^{itx}) =\int_0^\infty e^{itx}ae^{-ax} dx = a \cdot \frac{e}{it-a} |_0 ^ \infty $$

The integral is not hard to evaluate if you write it as $a\int_0^\infty e^{(it -a)x}~dx = \frac a{it - a}e^{(it - a)x}|_0^\infty$. I should add that that last expression should really be $\lim_{b \to \infty} \frac a{it - a}e^{(it - a)x}|_0^b$. Before going any further, there need to be some assumptions on the value of the expression $it - a$. You also, can't just "plug in" $\infty$ -- you need to work with a limit as shown above to evaluate the antiderivative.

 

[CivilSigma]

Makes sense. I assumed that it - a < 0 otherwise the integrand diverges to infintiy.