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hints10 - Mathematics 185 Intro to Complex Analysis Fall...

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Mathematics 185 – Intro to Complex Analysis Fall 2009 – M. Christ Hints for Problems from § 4.3 4.3.2 (i) For real x 6 = 0, sin 2 ( x ) x 2 is half of the real part of 1 - e 2 ix x 2 . Therefore we need to calculate half of the real part of R -∞ 1 - e 2 ix x 2 dx . (ii) This is the type of integral to which Prop 4.3.11 applies. To apply that Proposition, one needs to verify hypothesis (i) of Prop 4.3.6. We will discuss this issue in detail in class on Friday, so if you wish to skip this step for now, that is OK. (iii) I’ve slurred over a bothersome point: R -∞ 1 - e 2 ix x 2 dx is a triply improper integral. The integrals over [1 , ) and ( -∞ , - 1] converge absolutely since the absolute value of the integral does not exceed | x | - 2 . But the integral over [ - 1 , 1] is also improper. However, Z -∞ sin 2 ( x ) x 2 dx = lim ε 0 + Z | x |≥ ε sin 2 ( x ) x 2 dx = lim ε 0 + Z | x |≥ ε 1 2 Re 1 - e 2 ix x 2 dx = 1 2 Re lim ε 0 + Z | x |≥ ε 1 - e 2 ix x 2 dx .
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