https://file.aiquickdraw.com/mathbot/file/1727060234796xlpfkpqo.jpg

The problem given in the image can be summarized as follows:

Given that:

\[
i^{x + iy} = x + iy
\]

we need to prove that:

\[
x^2 + y^2 = e^{-y \pi}
\]

### Approach:

This expression involves complex exponentials and powers of \( i \), which can be analyzed using Euler's formula and properties of complex numbers. The steps to solve this would involve expressing the left-hand side in polar form, using properties of the imaginary unit \( i \), and comparing the real and imaginary parts.

Let’s break it down step-by-step.

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If i^{x + iy} = x + iy, then prove that x^2 + y^2 = e^{-y \pi}.

Explore the detailed solution to the problem involving complex numbers and exponential functions: If i^{x + iy} = x + iy, prove that x^2 + y^2 = e^{-y \pi}. This problem uses key concepts like Euler's formula and properties of complex numbers, suitable for undergraduate mathematics students.

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