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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2005 • Question 10

Question

Math 6210 • Real Analysis 2005 • Question 10

Determine the region of values of \(z\) for which \[ f(z)=\int _0^\infty \frac {e^{tz}}{1+t^2}\,dt \] defines a holomorphic function. (Hint: one approach is to use the holomorphicity criterion of Morera’s Theorem. If you pursue this approach, make certain to justify any application of Fubini’s theorem.)

Raw LaTeX

Determine the region of values of \(z\) for which
\[
f(z)=\int_0^\infty\frac{e^{tz}}{1+t^2}\,dt
\]
defines a holomorphic function. (Hint: one approach is to use the holomorphicity
criterion of Morera's Theorem. If you pursue this approach, make certain to
justify any application of Fubini's theorem.)

Topic Tags

Cauchy Integral Theory and Morera's Theorem Complex Integration and Real Integral Evaluation Holomorphic Functions and Cauchy-Riemann Equations Lebesgue Integration