Raising Complex Numbers to Powers of Angles - What Happens?

  • Context: Undergrad 
  • Thread starter Thread starter Andrea2
  • Start date Start date
  • Tags Tags
    Angles
Click For Summary
SUMMARY

Raising a complex number to the power of an angle in radians involves the use of Euler's formula. Specifically, for the complex number \(i\), the expression \((i)^a\) can be represented as \(e^{(a \cdot \log(i))}\), where \(\log(i) = i \cdot \frac{\pi}{2}\). This results in \((i)^a = e^{(i \cdot a \cdot \pi)}\), yielding a complex number that varies based on the angle \(a\). The discussion highlights the complexity of raising real numbers to angles compared to complex numbers.

PREREQUISITES
  • Understanding of complex numbers and their properties
  • Familiarity with Euler's formula and its applications
  • Knowledge of logarithmic functions in the context of complex analysis
  • Basic trigonometry, particularly in relation to angles in radians
NEXT STEPS
  • Study Euler's formula in depth, focusing on its implications for complex exponentiation
  • Explore the properties of logarithms in complex analysis, particularly \(\log(z)\)
  • Investigate the geometric interpretation of complex exponentiation on the complex plane
  • Learn about the applications of complex powers in fields such as electrical engineering and quantum mechanics
USEFUL FOR

Mathematicians, physics students, engineers, and anyone interested in the applications of complex numbers in advanced mathematics and engineering disciplines.

Andrea2
Messages
9
Reaction score
0
is there anyone who can tell me what happen when i raise a complex number to the power of an angle in rad? for example, what is the result of (i)^1rad?
 
Physics news on Phys.org
is there anyone who can tell me what happen when i raise a complex number to the power of an angle in rad? for example, what is the result of (i)^1rad?
Since you ask about the power of a COMPLEX number, so I suppose that you already know what happen when you raise a REAL number to the power of an angle in rad. For example, what is the result of 2^1rad ?
 
i don't know exactly what happen if i raise a real number to the power of an angle, i tried to answer also to this question, but five minutes ago i think I've found the answer to my first question, and i think that this problem is more difficult with real numbers than with complex! My solution is that (i)^a = e^(a x log(i)) with a=angle in rad. log(i) = i x p/2 (p=3.1416...), so (i)^a = e^(i x ap), that is a complex number...
 

Similar threads

Undergrad Formal definition of multiplication for real and complex numbers
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Constant raised to complex numbers
  • · Replies 14 ·
Replies
14
Views
2K
Undergrad Division of a complex number by zero
  • · Replies 7 ·
Replies
7
Views
5K
Undergrad About convergence of complex power series on the circle ##|z - z_0|=r##
  • · Replies 22 ·
Replies
22
Views
5K
Undergrad A beginner’s thoughts on rotation number -- Resource suggestion request
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Apparent counterexample to Cauchy-Goursat theorem (Complex Analysis)
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Intuitive understanding of Euler's identity?
  • · Replies 6 ·
Replies
6
Views
2K
Graduate A complicated integral to be evaluated by complex contour method
  • · Replies 14 ·
Replies
14
Views
4K
Undergrad Equivalence of two complex expressions
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Relation between a circle and line in complex plane
  • · Replies 16 ·
Replies
16
Views
3K