The expression \( e^{i\pi x} \) is closely related to Euler's formula, which states:

\[
e^{ix} = \cos(x) + i\sin(x)
\]

When \( \pi x \) is used as the argument, we get:

\[
e^{i\pi x} = \cos(\pi x) + i\sin(\pi x)
\]

This is a complex-valued function, meaning that its graph has two components:
1. The real part, \( \cos(\pi x) \)
2. The imaginary part, \( \sin(\pi x) \)

To graph \( e^{i\pi x} \), we will need to plot the real and imaginary components separately as functions of \( x \).

- **The real part \( \cos(\pi x) \)** will oscillate between -1 and 1 with period 2.
- **The imaginary part \( \sin(\pi x) \)** will also oscillate between -1 and 1 with period 2.

### Visualization Approach:
A common way to graph complex functions is by plotting both the real and imaginary parts as two separate curves on the same graph, or by using a 3D plot where one axis represents the real part, one represents the imaginary part, and the third represents \( x \).

Would you like me to generate a graph showing both the real and imaginary parts of this function?

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Here are 5 related questions to explore further:
1. What is Euler's formula and why is it important in complex analysis?
2. How do we interpret the magnitude and phase of a complex function like \( e^{i\pi x} \)?
3. What are some real-world applications of complex exponentials?
4. How does the periodicity of trigonometric functions affect the graph of \( e^{i\pi x} \)?
5. How can we extend this analysis to other complex functions, like \( e^{ix^2} \)?

**Tip:** Complex exponentials are key to understanding Fourier analysis, a powerful tool in signal processing and physics.

Explore how to graph the complex function e^{i\pi x} using Euler's formula. Learn how the real part (cos(\pi x)) and the imaginary part (sin(\pi x)) behave, with a focus on complex analysis and trigonometric functions. Suitable for undergraduate mathematics students.

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