Calculating Coherence Time: Understanding the Relationship Between g(t) and t_c

[Observer Two]

Homework Statement

I have the complex term g(t) = e^{\frac{-|t|}{t_c}} which is the degree of the coherence.

Homework Equations

Now I want to verify that:

t_c = \int_{-\infty}^\infty \! |g(t)|^2 \, dt

The Attempt at a Solution

\int_{-\infty}^\infty \! |g(t)|^2 \, dt = \int_{-\infty}^\infty \! |e^{\frac{-|t|}{t_c}}|^2 \, dt = \int_{-\infty}^\infty \! e^{\frac{-|t|}{t_c}} e^{\frac{|t|}{t_c}} \, dt = \int_{-\infty}^\infty \! 1 \, dt

2 Problems now.

First: The integral doesn't have a value if I integrate from - infinity to infinity.
Second: The value of the indefinite integral is t. Not t_c.

What am I missing here?

 

[NasuSama]

You didn't multiply $e^{-\frac{|t|}{t_c}}$ by itself. Instead, the second multiplier misses the negative sign. Check your work carefully and try again evaluating the integral.

 

[Observer Two]

Huh? I'm really missing something here.

|z|^2 = z z^*

So if in my case z = e^{\frac{-|t|}{t_c}} then

z^* = e^{\frac{|t|}{t_c}}

Or not?