complex number, absolute value of a simple function

[tommycashmoney]

this is my function and i want to apply plancherel this formula

i know that the result is

i don't understand how we end up with e^-8abs(w) in the bracket, can anybody show me the steps between the last two expressions?

 

[Deleted member 4993]

View attachment 24933
this is my function and i want to apply plancherel View attachment 24934 this formula

i know that the result is View attachment 24935

i don't understand how we end up with e^-8abs(w) in the bracket, can anybody show me the steps between the last two expressions?

Did you calculate:

|g(w)| and |g(w)|² ?​

​

- what did you get?​

 

[Dr.Peterson]

View attachment 24933
this is my function and i want to apply plancherel View attachment 24934 this formula

i know that the result is View attachment 24935

i don't understand how we end up with e^-8abs(w) in the bracket, can anybody show me the steps between the last two expressions?

Multiply [MATH]\hat{g}(\omega)[/MATH] by its conjugate to get [MATH]|\hat{g}(\omega)|^2[/MATH].

 

[tommycashmoney]

why do i have to multiply it by its conjugate and not by itself?

 

[Deleted member 4993]

why do i have to multiply it by its conjugate and not by itself?

That IS the RULE of complex numbers!!

 

[Dr.Peterson]

why do i have to multiply it by its conjugate and not by itself?

See equation (10) here. You may find the rest useful to read, too.

 

[HallsofIvy]

We want the "absolute value" of a number, even a complex number, to be a non-negative real numbe. If you square a complex number the result is NOT in general a real number so the square root is not in general a real number. In order to get a non-negative real number we need to multiply the number, a+ bi, by its "complex conjugate", a- bi.