# Characteristic Function Integrand Evaluation

CivilSigma

## Homework Statement

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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.

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Homework Helper
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## 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.

Last edited by a moderator:
• CivilSigma
Mentor
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
CivilSigma
Makes sense. I assumed that it - a < 0 otherwise the integrand diverges to infintiy.