Lecture3 - ESE 502 Mathematics of Modern Engineering II...

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ESE 502 Mathematics of Modern Engineering II Lecture 3 ”Extending the Fourier series: Fourier integral”. What we know: Any function f defined on a finite interval , without loss of generality we can assume it to be [ L, L ], and satisfying the Existence conditions of the Fourier seris, admits the representation f ( x ) = a 0 + k =1 [ a k cos L x + b k sin L x ] where a 0 , a k and b k , k = 1 , 2 , ... are the corresponding Fourier coefficients. When we introduce the notation w k := L , then the sequence of points w 0 , w 1 , w 2 , ... will define on the interval [0 , ) a partition with the diameter Δ w = w k 1 w k = π L In new notations, the representation above is written as f ( x ) = 1 2 L L L f ( y ) dy + 1 π k =1 [(cos w k x w L L f ( y )(cos w k y ) dy +(sin w k x w L L f ( y )(sin w k y ) dy ] . Now: We are interested to see what happens on the right side on the last identity if we take the limit L → ∞ ? It is not hard to expect that - given the convergence on the right side will be warrantied - the limiting identity will become f ( x ) = 1 π 0 [cos wx −∞ f ( y ) cos wydy + sin wx −∞ f ( y ) sin wydy ] dw, 1
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or, in other notations: f ( x ) = 0 [ A ( w ) cos wx + B ( w ) sin wx ] dw (1) where A ( w ) = 1 π −∞ f ( y ) cos wydy, (2) B ( w ) = 1 π −∞ f ( y ) sin wydy. (3) The above formulas (1)-(3) determine the Fourier integral representation of the function f .
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  • Fall '09
  • Periodic function, Mathematical analysis, Leonhard Euler

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