Does e^z - z^2 = 0 Have Infinite Solutions?

Click For Summary
SUMMARY

The equation ez - z2 = 0 has infinitely many solutions due to the periodic nature of the exponential function ez. By rewriting the equation as z2 = e2 log z, we derive the form e{z - 2 log z} = 1. This leads to the conclusion that the solutions can be expressed as z - 2 log z = 2πn, where n is an integer. The Lambert W function is identified as a useful tool for solving this equation.

PREREQUISITES
  • Understanding of complex analysis, particularly the properties of the exponential function.
  • Familiarity with logarithmic functions and their branches.
  • Knowledge of the Lambert W function and its applications.
  • Basic skills in solving transcendental equations.
NEXT STEPS
  • Study the properties of the Lambert W function and its applications in solving equations.
  • Research the periodicity of the exponential function and its implications in complex analysis.
  • Explore advanced techniques for solving transcendental equations, focusing on logarithmic transformations.
  • Learn about fixed branches of logarithmic functions and their significance in complex equations.
USEFUL FOR

Mathematicians, students of complex analysis, and anyone interested in solving transcendental equations involving exponential and logarithmic functions.

Likemath2014
Messages
17
Reaction score
0
How can we show that the following equation has infinitely many solutions
e^z-z^2=0.
Thanks
 
Physics news on Phys.org
Work within a fixed branch of logz and write : ## z^2=e^{2logz} ##

Once you find a solution, you have infinitely-many, by periodicity of ##e^z##.
 
Continuing WWGD's post: you get ##e^z=e^{2\log z}## and thus the equation ##e^{z-2log z}=1##. In other words you are looking at the solutions of each of the equations ##z-2\log z=2\pi n## for ##n\in\mathbb{Z}##.
 
It seems that something like Lambert's W function may be helpful in finding solutions to platetheduke's equation.
 

Similar threads

Undergrad Bounding modulus of complex logarithm times complex power function
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Evaluate two complex integrals along a parabolic contour
  • · Replies 2 ·
Replies
2
Views
972
Undergrad Apparent counterexample to Cauchy-Goursat theorem (Complex Analysis)
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Question about branch of logarithms
  • · Replies 14 ·
Replies
14
Views
4K
Undergrad About convergence of complex power series on the circle ##|z - z_0|=r##
  • · Replies 22 ·
Replies
22
Views
4K
MHB Solving a complex value equation
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad I need integrals Int[0,infty]t^(a-1) e^(-t) F(z, e^(-t))dt
  • · Replies 4 ·
Replies
4
Views
4K
MHB How Do You Calculate the Real and Imaginary Parts of \( e^{e^z} \)?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Calculate the residue of 1/Cosh(pi.z) at z = i/2 (complex analysis)
  • · Replies 7 ·
Replies
7
Views
4K
MHB Complex Analysis: Does $\int_{C(0,10)} f(z)$ Equal 0?
  • · Replies 2 ·
Replies
2
Views
2K