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While playing around with a plotting software, i just found out that

$f(x) = i^x = \cos(x·\frac{\pi}{2})$

  1. How does this connect to Euler's formula?
  2. Obviously, here, the alternating sign change is responsible for periodicity and form of the cosine. Is this also true for Euler's formula?

Please don't beat me, i'm an engineering student.

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    The thing you have to watch out for when you do this is that not only $i=e^{i\pi/2}$, but also $i=e^{5i\pi/2}$, $i=e^{9i\pi/2}$, and so on. It's tempting to say that $i^i = e^{i\cdot i\pi/2} = e^{-\pi/2} \approx 0.21$, which is correct, but only as far as it goes, since also $i^i = e^{i\cdot 5i\pi/2} = e^{-5\pi/2} \approx 0.0004$. Really the exponentiation operator is not uniquely defined on the complex numbers.2012-05-28

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The quantity $i^x$ by itself is not well-defined. The way one would like to define it is $i^x = e^{x\log i}$, and then use the Taylor series for the exponential to compute $e^{x\log i}$. The problem with this is that $\log i$ is not well-defined: there are infinitely many possible values of $\log i$, namely $\log i = \frac{\pi i}{2} + 2\pi in$ for any $n\in \mathbb{Z}$. Thus to define $i^x$, you have to make a choice as to which one of these logarithms you are using. The standard choice would be $\log i = \pi i/2$. In this case, $i^x = e^{x\log i} = e^{i\pi x/2} = \cos(\pi x/2) + i\sin(\pi x/2).$ However, if you had chosen $\log i = \pi i/2 + 2\pi in$ for some $n\neq 0$, then $i^x = e^{x(\pi i/2 + 2\pi in)} = \cos(\pi x/2 + 2\pi nx) + i\sin(\pi x/2 + 2\pi nx).$

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My answer is not about your questions, which are wonderfully addressed in the answer and comments before me, but about your discovery. This should be a good warning: a lot of computer systems plot only the real part of a given expression. For example, it is true for Maple. Therefore, students should be careful because sometimes what is shown in the plot is not exactly in the original expression.

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    @bijan No, a complex valued function is a mapping of plane to plane. There are different ways to visualize such mappings, e.g., plotting $\Re (f(z))$ or $|f(z)|$.2012-05-28