11.7 Fourier Integral
Transcript of 11.7 Fourier Integral
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 0∞ sin 𝑤
𝑤𝑑𝑤.
Example: Find the Fourier integral of the following function.
𝑓 𝑥 = 𝑒−𝑥 𝑥 > 00 𝑥 < 0
Lecture 7:
Recall that
Recall
Thus,
𝑓𝑐(𝑤)
𝑓𝑐(𝑤)
𝑓(𝑥)
𝑓𝑐 𝑤 =2
𝜋 0
∞
𝑓 𝑥 cos𝑤𝑥 𝑑𝑥
𝑓𝑐(𝑤)
𝑓 𝑥 =2
𝜋 0
∞ 𝑓𝑐 𝑤 cos𝑤𝑥 𝑑𝑤
The Fourier cosine transform of 𝑓(𝑥)
The inverse Fourier cosine transform of 𝑓𝑐(𝑤)
𝑓(𝑥)
𝑓𝑠 𝑤 =2
𝜋 0
∞
𝑓 𝑥 sin𝑤𝑥 𝑑𝑥
𝑓𝑠(𝑤)
𝑓 𝑥 =2
𝜋 0
∞ 𝑓𝑠 𝑤 sin𝑤𝑥 𝑑𝑤
The Fourier sine transform of 𝑓(𝑥)
The inverse Fourier sine transform of 𝑓𝑠(𝑤)
Similarly, for an odd function the Fourier sine transform and the inverse Fourier sinetransform of 𝑓 𝑥 are defined as follows.
Other notions are
Exercise: By integration by parts an recursion find ℱ𝑐 𝑒−𝑥 .
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 𝑓 𝑥 = 𝑒−𝑎𝑥 , where 𝑎 > 0.
Lecture 9: proof of Relation (2)