I came across this question (attached below). I tried to solve it but got stuck in the portion where we need to calculate the inverse tranform. I found a solution of this question (also attached below) but there, in the encircled portion, I couldn't get how they took a 2x factor out and change the limits from (-∞, +∞) to [0,+∞ ). I know that we can do this changing of limits only if it's an even function and can take the limiting points [0,+∞ ) iff the function is of the form y = ax2 . But here the term inside the exponential function is of the form y = ax2 + bx + cix, where i =complex number, and accordingly the limit should change to some random [m,+∞ ) in place of [0,+∞ ). Also, the 2x factor would not be there because the limiting point is changing from [0,+∞ ) to [m,+∞ ) and the graph will not be symmetrical across X=0 Axis.
I will be highly grateful if you can kindly clarify my doubt or let me know where I am making a mistake in understanding the question.
You should be careful to notice that the exp argument in the C(z) integral (the row just above the one you circlesd) contains an "absolute value" of the frequency variable... b|w| .
The steps that brings you from the 1st line to the circled line are:
1. separate the integral into the sum of an integral over negative w range (so that b|w|=-bw) and an integral over positive w (where b!w|=bw).
2. transform w --> -w into the first integral so to make it running from 0 to infinity (as the second integral)
3. recover a unique integral from 0 to infinity... you will notice the appearance of a sum like exp(ciw)+exp(-ciw), which is 2Re(exp(icw))
4. Use the property of the complex numbers that Re(az)=aRe(z) if a is real and z is complex.
you will obtain the circled expression correctly.
Giuseppe Curro Thank you very much Sir for helping me understand the point where I was making the mistake.
- Badges
- Science topic
- Similar topics
- Physics