Complex Exponential: What is e^jw∞?

Click For Summary
SUMMARY

The expression e^{jw∞} cannot be defined within the complex plane due to the fact that infinity (∞) is not a complex number. To explore this concept, one may consider the Riemann sphere, which allows for a different perspective on complex analysis. The function e^z requires a specific branch of log(z) to be well-defined and single-valued; however, since log(z) is not defined across the entire complex plane, it cannot be considered analytic in the extended complex plane.

PREREQUISITES
  • Understanding of complex analysis concepts
  • Familiarity with the Riemann sphere
  • Knowledge of the logarithm function in complex analysis
  • Basic principles of analytic functions
NEXT STEPS
  • Study the properties of the Riemann sphere in complex analysis
  • Learn about branches of the logarithm function in complex variables
  • Explore analytic continuation and its applications
  • Investigate the implications of infinity in complex functions
USEFUL FOR

Mathematicians, students of complex analysis, and anyone interested in advanced topics related to complex functions and their definitions.

tommyhakinen
Messages
34
Reaction score
0
what is e^{jw\infty} ?
 
Physics news on Phys.org
You cannot define it in the complex plane, since oo is not a complex number. You may

want to work on the Riemann sphere, or something, if you want to work with the log,

tho I don't see how to do so at this point. e^z must be defined in terms of a branch

of logz , to be well-defined, in the sense of being single-valued, among other things.

Since logz is not even defined in the whole complex plane, you cannot define a logz

to be analytic in the extended complex plane
 

Similar threads

Graduate A complicated integral to be evaluated by complex contour method
  • · Replies 14 ·
Replies
14
Views
3K
MHB How Do You Calculate the Real and Imaginary Parts of \( e^{e^z} \)?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Why is this function not ##L^1(\mathbb{R} \times \mathbb{R})##?
  • · Replies 6 ·
Replies
6
Views
3K
High School Getting from complex domain to real domain
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Complex exponential to a power
  • · Replies 4 ·
Replies
4
Views
1K
Worth learning complex exponential trig derivations in precalc?
  • · Replies 22 ·
Replies
22
Views
3K
Undergrad Formal definition of multiplication for real and complex numbers
  • · Replies 2 ·
Replies
2
Views
2K
High School Mistake in my complex exponentiation: where?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Connection between absolute continuity of function and measure
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Polynomial vs. Exponential time
  • · Replies 2 ·
Replies
2
Views
743