Calculating (a+bi)^(c+di): How to Find it?

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Discussion Overview

The discussion centers around the calculation of the expression (a+bi)^(c+di), exploring methods to evaluate it, particularly through the use of exponentials and logarithms of complex numbers. Participants delve into theoretical approaches and mathematical identities relevant to complex exponentiation.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant asks how to find (a+bi)^(c+di) and suggests breaking it down into [(a+bi)^c]x[(a+bi)^di], questioning how to handle the (a+bi)^di part.
  • Another participant introduces identities involving exponentials and logarithms of complex numbers, providing formulas that could aid in the calculation.
  • A participant expresses confusion regarding the initial equations provided, indicating a lack of understanding of exponentials and logarithms of complex numbers.
  • One participant proposes a method to express (a+bi)^(di) using polar coordinates and De Moivre's theorem, detailing the transformation into exponential form.
  • Another participant questions whether there are alternative methods to solve the problem without using exponentials or logarithms.
  • A participant reiterates their confusion about the initial equations and emphasizes the necessity of understanding complex exponentiation concepts to tackle such problems.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the mathematical concepts involved, with some agreeing on the utility of exponentials and logarithms while others remain uncertain or seek alternative methods. The discussion does not reach a consensus on a single approach to the problem.

Contextual Notes

Some participants note the multi-valued nature of the logarithm function, which may complicate the discussion. There are also indications of missing foundational knowledge among some participants regarding complex numbers and their properties.

kishtik
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What is (a+bi)^(c+di) ? How can I find this?
=[(a+bi)^c]x[(a+bi)^di]=? Now I can go binomial for the first part but what about (a+bi)^di?
 
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Have you dealt with exponentials and logarithms of complex numbers yet?

The identities that'll help you are:

[tex] \begin{array}{l}<br /> \forall x,y \in {\rm R} \\ <br /> \exp (x + iy) = \exp (x)(\cos (y) + i\sin (y)) \\ <br /> \log _e (x + iy) = \log _e (\sqrt {x^2 + y^2 } ) + i\arctan (y/x) \\ <br /> \end{array}[/tex]

(I've glossed over the fact that the log function is actually multi-valued ... let me know if you need this explained further).

[tex] \begin{array}{l}<br /> \forall z,w \in {\rm C }, w \neq 0 \\ <br /> \log _e (z^w ) = w\log _e (z) \\ <br /> z^w = \exp (w\log _e (z)) \\ <br /> \end{array}[/tex]

See how you get on.
 
I couldn't understand the first three quations although I did the last three. And I have no idea about expotentials and logarithms of complex numbers. Thanks.
 
(a+ib)^{di} = ((m \exp (ni))^d)^i
m = \sqrt{a^2 + b^2}, n = arctan (b/a)
= (m \exp(ni))^i)^d
= ((m^i) \exp(-n))^d
= m^{id} \exp(-nd)
m^{id} is a complex number.



Regarding the equations,
They are pretty simple
The first equation is the famous De-Moivre's theorem. Prrof can be found in any algebra book.
For Eqn 2, from De-Moivre's theorem,
[tex] \begin{array}{l}<br /> x + iy = \sqrt(x^2+y^2)\exp(i\arctan(y/x))<br /> \end{array}[/tex]
Take logarithms on both sides and you get equation 2.
 
I'm not sure if there are ways to solve this without exp, log etc.

Does anyone know another way?
 
kishtik said:
I couldn't understand the first three quations although I did the last three. And I have no idea about expotentials and logarithms of complex numbers. Thanks.
Yeah, you really need to look up exponents, logs, trigonometric functions and hyperbolic functions in relationship to complex numbers otherwise you'll struggle to deal with such problems.
 

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