SOLUTION: I am looking for a general solution to a complex number raised to a complex power. I recall that Euler had a solution, but I can't recall what it was. However, it was in the form o
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Question 300834: I am looking for a general solution to a complex number raised to a complex power. I recall that Euler had a solution, but I can't recall what it was. However, it was in the form of (something here) + i (something here), and used logarithms heavily (as is an Euler trademark). Any help will be greatly appreciated. Thanks.
Found 2 solutions by solver91311, Edwin McCravy:
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
To calculate ^{c\ +\ di})
:
Let \ =\ \pm\arctan\frac{b}{a})
taking the sign appropriately so that )
Let )
Then ^{c\ +\ di} =\ \left(a^2\,+\,b^2\right)^{c/2}\,\cdot\,e^{-d\varphi}\,\cdot\,\left(\cos\theta\ +\ i\sin\theta\right))
John
Answer by Edwin McCravy(20062) (Show Source): You can put this solution on YOUR website!
Euler's equation is:
So
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