11.7 fourier integral
Post on 20-Apr-2022
17 Views
Preview:
TRANSCRIPT
11.7 Fourier Integral
As an aim of this section we want to solve this problem
Recall that
THEOREM 1 (Fourier Integral)
Using this, evaluate 0β sin π€
π€ππ€.
Example: Find the Fourier integral of the following function.
π π₯ = πβπ₯ π₯ > 00 π₯ < 0
Lecture 7:
Recall that
Recall
Thus,
ππ(π€)
ππ(π€)
π(π₯)
ππ π€ =2
π 0
β
π π₯ cosπ€π₯ ππ₯
ππ(π€)
π π₯ =2
π 0
β ππ π€ cosπ€π₯ ππ€
The Fourier cosine transform of π(π₯)
The inverse Fourier cosine transform of ππ(π€)
π(π₯)
ππ π€ =2
π 0
β
π π₯ sinπ€π₯ ππ₯
ππ (π€)
π π₯ =2
π 0
β ππ π€ sinπ€π₯ ππ€
The Fourier sine transform of π(π₯)
The inverse Fourier sine transform of ππ (π€)
Similarly, for an odd function the Fourier sine transform and the inverse Fourier sinetransform of π π₯ are defined as follows.
Other notions are
Exercise: By integration by parts an recursion find β±π πβπ₯ .
Linearity of sine and cosine transforms
Similarly,
Lecture 8: Prove the Relations 4a, 4b, 5a and 5b and also solution of Problems 12 and 13 of 11.8
Exercise: Find the Fourier sine transform of π π₯ = πβππ₯ , where π > 0.
Lecture 9: proof of Relation (2)
top related