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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2008 • Question 8

Question

Math 6210 • Real Analysis Fall 2008 • Question 8

Let \(\alpha >1\) be arbitrary. Show that the equation \[ \alpha -z-e^{-z}=0 \] has exactly one solution in the half plane \(\{z:\operatorname {Re}z>0\}\), and moreover, this solution is real.

Raw LaTeX

Let \(\alpha>1\) be arbitrary. Show that the equation
\[
\alpha-z-e^{-z}=0
\]
has exactly one solution in the half plane
\(\{z:\operatorname{Re}z>0\}\), and moreover, this solution is real.

Topic Tags

Argument Principle and Rouche's Theorem