11.7 fourier integral
TRANSCRIPT
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11.7 Fourier Integral
As an aim of this section we want to solve this problem
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Recall that
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THEOREM 1 (Fourier Integral)
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Using this, evaluate 0∞ sin 𝑤
𝑤𝑑𝑤.
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Example: Find the Fourier integral of the following function.
𝑓 𝑥 = 𝑒−𝑥 𝑥 > 00 𝑥 < 0
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Lecture 7:
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Recall that
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Recall
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Thus,
𝑓𝑐(𝑤)
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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)
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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
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Exercise: By integration by parts an recursion find ℱ𝑐 𝑒−𝑥 .
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Linearity of sine and cosine transforms
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Similarly,
Lecture 8: Prove the Relations 4a, 4b, 5a and 5b and also solution of Problems 12 and 13 of 11.8
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Exercise: Find the Fourier sine transform of 𝑓 𝑥 = 𝑒−𝑎𝑥 , where 𝑎 > 0.
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Lecture 9: proof of Relation (2)
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