Complex Exponential: Why is e Used?

Click For Summary
SUMMARY

The discussion clarifies that the equation e^{ix} = cos(x) + i sin(x) is not merely a definition but fundamentally relies on the mathematical properties of the number e as the base of the exponential function. This relationship is derived from Taylor's series for e^x, cos x, and sin x, where the unique property of e, being its own derivative, simplifies the derivation. Additionally, it emphasizes the necessity of using radian measure for the arguments in sine and cosine functions to maintain the validity of the equation.

PREREQUISITES
  • Understanding of Euler's formula and its implications
  • Familiarity with Taylor series expansions
  • Knowledge of derivatives and their applications
  • Basic trigonometry, specifically radian measure
NEXT STEPS
  • Study the derivation of Euler's formula using Taylor series
  • Explore the properties of the exponential function e^x and its derivatives
  • Learn about the significance of radian measure in trigonometric functions
  • Investigate applications of complex numbers in engineering and physics
USEFUL FOR

Mathematicians, physics students, and anyone interested in the applications of complex numbers and exponential functions in advanced mathematics.

Repetit
Messages
128
Reaction score
2
Hey!

I was wondering, is it merely a definition that

[tex]e^{ix}= cos(x) + i sin(x)[/tex]

or is it actually important that it is the number e which is used as base for the exponential?

Thanks!
 
Physics news on Phys.org
Have a look at http://en.wikipedia.org/wiki/Euler%27s_formula" :smile:
 
Last edited by a moderator:
No, it's not a definition, yes, it is important the "e" be the base of the exponentials. That formula can be derived from the Taylor's series for ex, cos x, and sin x. Taylor's series depends upon the derivatives and the derivative of ex happens to be ex itself. The derivative of ax is axln(a) so corresponding formulas are more complicated.
For similar reasons, it is also important that the argument in the sine and cosine be interpreted as radian measure, not degrees.
 

Similar threads

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
MHB Solving a complex value equation
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Formal definition of multiplication for real and complex numbers
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Connection between absolute continuity of function and measure
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Evaluate two complex integrals along a parabolic contour
  • · Replies 2 ·
Replies
2
Views
1K
Worth learning complex exponential trig derivations in precalc?
  • · Replies 22 ·
Replies
22
Views
3K
High School Computing a tricky complex math problem - where did I go wrong?
  • · Replies 7 ·
Replies
7
Views
1K
Undergrad Bounding modulus of complex logarithm times complex power function
  • · Replies 4 ·
Replies
4
Views
4K
Is ##f(x)=2^{x}-1## considered an exponential function?
  • · Replies 12 ·
Replies
12
Views
3K