Complex solution to cos x = 2?

  • Context: Graduate 
  • Thread starter Thread starter johann1301
  • Start date Start date
  • Tags Tags
    Complex Cos
Click For Summary
SUMMARY

Complex solutions to the equation cos x = 2 exist by utilizing an alternate definition of the cosine function: cos(x) = (e^(i x) + e^(-i x))/2. By setting the right side equal to 2, one can derive a quadratic equation in terms of e^x. This approach leads to the identification of complex solutions for x, confirming that such solutions are indeed valid within the realm of complex analysis.

PREREQUISITES
  • Understanding of complex numbers and their properties
  • Familiarity with the exponential function and Euler's formula
  • Knowledge of quadratic equations and their solutions
  • Basic grasp of trigonometric functions and their definitions
NEXT STEPS
  • Study the derivation of Euler's formula and its applications in complex analysis
  • Learn how to solve quadratic equations involving exponential functions
  • Explore the implications of complex solutions in trigonometric equations
  • Investigate the broader context of complex functions and their graphical representations
USEFUL FOR

Mathematicians, physics students, and anyone interested in complex analysis and solving trigonometric equations in the complex plane.

johann1301
Messages
216
Reaction score
1
Are there any complex solutions to cos x = 2?
 
Physics news on Phys.org
Yes.
 
Now for the next question. How does one find them, right? Well, you just need to take an alternate definition for the cosine function:

cos(x) = \frac{e^{i x} + e^{- i x}}{2}

Set the right side equal to 2, and solve for x. Hint, it's going to involve a quadratic equation in e^x.
 
Thanks!
 

Similar threads

Undergrad How to see that cos(z) is unbounded, but cos(xy)+isin(xy) is bounded?
  • · Replies 8 ·
Replies
8
Views
2K
High School Principal square root of a complex number, why is it unique?
  • · Replies 13 ·
Replies
13
Views
6K
High School Getting from complex domain to real domain
  • · Replies 3 ·
Replies
3
Views
2K
MHB Solving a complex value equation
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Even with a whimsical mathematical usage, solutions are obtained!
  • · Replies 5 ·
Replies
5
Views
2K
Insights Views On Complex Numbers
  • · Replies 108 ·
4
Replies
108
Views
11K
How do we write a sinusoidal solution to a 2nd order DE as a sum of exponentials raised to complex roots?
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Evaluate two complex integrals along a parabolic contour
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad ##1^x =2## has complex solutions?
  • · Replies 7 ·
Replies
7
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