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)