Math Problem Statement

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

Solution

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

Given that:

ix+iy=x+iyi^{x + iy} = x + iy

we need to prove that:

x2+y2=e−yπx^2 + y^2 = e^{-y \pi}

Approach:

This expression involves complex exponentials and powers of ii, 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 ii, and comparing the real and imaginary parts.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Exponential Functions
Euler's Formula

Formulas

Euler's Formula: e^{iθ} = cos(θ) + i sin(θ)
Magnitude of Complex Numbers: |z| = sqrt(x^2 + y^2)

Theorems

Euler's Theorem
Properties of the Imaginary Unit

Suitable Grade Level

Undergraduate Mathematics

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