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sheet2 - Mathematical Methods I Dr Gordon Ogilvie Example...

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Mathematical Methods I Natural Sciences Tripos, Part IB Dr Gordon Ogilvie Michaelmas Term 2008 Example Sheet 2 1. Define δ ǫ ( x ) for ǫ > 0 by δ ǫ ( x ) = 1 πx sin parenleftBig x ǫ parenrightBig . (a) Evaluate integraldisplay −∞ δ ǫ ( x ) d x given that integraldisplay 0 sin x x d x = π 2 . (b) Argue that for a ‘good’ function f and a constant ξ lim ǫ 0+ integraldisplay −∞ δ ǫ ( x ξ ) f ( x ) d x = f ( ξ ) . (c) Sketch δ ǫ ( x ) and comment. 2. (a) Starting from the definition that δ ( x ) is the generalized function such that for all ‘good’ functions f ( x ) integraldisplay −∞ δ ( x ξ ) f ( x ) d x = f ( ξ ) , show that, for constant a negationslash = 0, ( x ) = 0 and δ ( ax ) = 1 | a | δ ( x ) . (b) Evaluate integraldisplay −∞ | x | δ ( x 2 a 2 ) d x , where a is a non-zero constant. Hint: the answer is not 2 a . If keen, discuss the case a = 0. 3. The differential equation y ′′ + y = H ( x ) H ( x ǫ ) , where H is the Heaviside step function and ǫ is a positive parameter, represents a simple harmonic oscillator subject to a constant force for a finite time. By solving 1
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the equation in the three intervals of x
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