Another question about math equality

Thread Starter

PsySc0rpi0n

Joined Mar 4, 2014
1,446
I'll use this thread to ask other math realted questions such as:

I have a property, in the context of Fourier Transform properties that says:

\[\delta \left ( a\cdot t \right ) = \frac{1}{a}\cdot \delta \left ( t \right )\]

and I need to apply it to the following expresison:
\[\delta \left ( \frac{\omega }{2} -1\right )\]

I say the result should be:
\[\begin{matrix} 1\rightarrow \delta \left ( \frac{\omega }{2} -1\right )\\ \\ 2\rightarrow \delta \left ( \frac{1}{2} \cdot \left ( \omega -2 \right )\right )\\ \\ 3\rightarrow 2\cdot \delta \left ( \omega -2 \right ) \end{matrix}
\]

where at:

\[2\]

I consider my 'a' to be:
\[\frac{1}{2}\]

and my 't' to be
\[\omega - 2\]

after looking to the left side of the property stated at the beginning.

My teacher wrote in the whiteboard that the result is:

\[\frac{1}{2}\delta \left ( \omega -2\right )\]

Am I wrong?
 

WBahn

Joined Mar 31, 2012
24,684
Using a single thread to ask different questions is a recipe for chaos. You ARE going to get responses to the different questions, without any indication of which question is being responded to, all intermixed.
 

Thread Starter

PsySc0rpi0n

Joined Mar 4, 2014
1,446
Using a single thread to ask different questions is a recipe for chaos. You ARE going to get responses to the different questions, without any indication of which question is being responded to, all intermixed.

Ok, so I'll quit this one.

And about the last question I asked, I think I'm correct and teacher might have made a mistake there!
 

WBahn

Joined Mar 31, 2012
24,684
I think you are starting off on a bad footing. Where are you getting that "property" from? Please provide a reference, if you can.

I suspect you are confusing it with a common definition of the delta function in which the delta function is the following function in the limit that ε goes to zero.

\[
\delta \( t-a \) \; = \; \frac{1}{\epsilon}
\]

for |t-a| < ε/2 and 0 otherwise.
 

Thread Starter

PsySc0rpi0n

Joined Mar 4, 2014
1,446
I think you are starting off on a bad footing. Where are you getting that "property" from? Please provide a reference, if you can.

I suspect you are confusing it with a common definition of the delta function in which the delta function is the following function in the limit that ε goes to zero.

\[\delta \( t-a \) \; = \; \frac{1}{\epsilon}
\]

for |t-a| < ε/2 and 0 otherwise.
I'm trying to find the property in Google but still no luck. Teacher told us about it in a class! Just wrote it in the whiteboard.


PS: Your 'tex' tags are not working. Don't hit enter after the starting/opening 'tex' tag. Just let it be int he same line as the tag! It works for me!
 
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Thread Starter

PsySc0rpi0n

Joined Mar 4, 2014
1,446
Ok, I found it!

\[\displaystyle{\int_{-\infty}^{+\infty}\delta \left ( \alpha \cdot x \right )\,dx = \int_{-\infty}^{+\infty}\delta \left ( u \right )\cdot \frac{du}{\left | \alpha \right |}=\frac{1}{\left | \alpha \right |}}\]

and so,

\[\displaystyle{\delta \left ( \alpha \cdot x \right ) = \frac{1}{\left | \alpha \right |}\cdot \delta \left ( x \right )}\]


From Wikipedia,
https://en.wikipedia.org/wiki/Dirac_delta_function#Scaling_and_symmetry
 
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