Images under exp in complex plane

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SUMMARY

The discussion focuses on finding the image of the line defined by the equation y = x under the exponential function exp(z) in the complex plane. Participants suggest starting with the parametric form x = t and y = t, leading to the expression exp(t + it) = exp(t)(cos(t) + i sin(t)). As t approaches infinity, the image spirals outward, while as t approaches negative infinity, the image converges towards the origin. This analysis highlights the behavior of complex exponential functions in relation to linear paths in the complex plane.

PREREQUISITES
  • Understanding of complex numbers and the complex plane
  • Familiarity with the exponential function exp(z) and Euler's formula
  • Knowledge of parametric equations and their graphical representation
  • Basic concepts of limits in calculus
NEXT STEPS
  • Explore the properties of complex exponential functions and their transformations
  • Learn about polar coordinates and their application in complex analysis
  • Study the behavior of complex functions as they approach infinity
  • Investigate the graphical representation of parametric equations in the complex plane
USEFUL FOR

Mathematicians, physics students, and anyone studying complex analysis or exploring the behavior of functions in the complex plane.

erraticimpulse
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Describe the image under exp of the line with equation y = x. To do this you should find an equation (at least parametrically) for the image (you can start with the parametric form x = t; y = t), plot it reasonably carefully, and explain what happens in the limits as t approaches infinity and t approaches negative infinity.

Not very certain where I should start. Should I use rectangular coordinates or polar?
 
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Here exp is notation for euler's number in the complex plane.
 
You have the line in parametric form, are you able to write down it's image under exp(z) in parametric form?
 

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