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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2018 • Question 8

Question

Math 6420 • PDE Fall 2018 • Question 8

Calculate the following integrals (\(a\) and \(b\) are positive parameters) \[ \begin {array}{rcl} \text {(a)}\quad I(a)&=&\displaystyle \int _{-\infty }^{\infty }\frac {\sin ax}{x}\,dx,\\[2.2ex] \text {(b)}\quad I(a,b)&=&\displaystyle \int _{-\infty }^{\infty } \frac {\cos ax-\cos bx}{x^2}\,dx. \end {array} \]

Raw LaTeX

Calculate the following integrals (\(a\) and \(b\) are positive parameters)
\[
\begin{array}{rcl}
\text{(a)}\quad I(a)&=&\displaystyle\int_{-\infty}^{\infty}\frac{\sin ax}{x}\,dx,\\[2.2ex]
\text{(b)}\quad I(a,b)&=&\displaystyle\int_{-\infty}^{\infty}
\frac{\cos ax-\cos bx}{x^2}\,dx.
\end{array}
\]

Topic Tags

Cauchy Integral Formula and Integral Theorems Fourier Transform and Sampling Theory Linear Programming and Optimization Real Integral Evaluation by Complex Methods