Math Problem Statement

simplify this image

Simplify this expression with complex exponentials.

Solution

The expression simplifies to:

14((e4iπt+1)(e2iπt+eiπ(60t+16)+eiπ(66t+13))eiπ(60t+16)+2eiπ(68t+12)+2eiπ(84t+16))eiπ(72t+13)\frac{1}{4} \left( (e^{4i\pi t} + 1)(e^{2i\pi t} + e^{i\pi\left(\frac{60t + 1}{6}\right)} + e^{i\pi\left(\frac{66t + 1}{3}\right)})e^{i\pi\left(\frac{60t + 1}{6}\right)} + 2e^{i\pi\left(\frac{68t + 1}{2}\right)} + 2e^{i\pi\left(\frac{84t + 1}{6}\right)} \right) e^{-i\pi\left(\frac{72t + 1}{3}\right)}

This is a more compact representation of the original complex exponential terms.

Would you like further details or clarification on how this was derived?

Here are 5 related questions:

  1. How can Euler's formula be applied to further simplify this expression?
  2. What does each exponential term represent in terms of sinusoidal functions?
  3. How would this expression behave as tt approaches specific values like 0 or 1?
  4. Can the frequency components of the expression be interpreted in terms of Fourier series?
  5. How does the combination of real and imaginary parts affect the overall function?

Tip: When simplifying complex exponentials, always consider Euler's formula eix=cos(x)+isin(x)e^{ix} = \cos(x) + i\sin(x) to convert them into trigonometric forms for easier interpretation.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Euler's Formula
Exponential Functions
Trigonometry

Formulas

Euler's Formula: e^{ix} = cos(x) + i sin(x)
Properties of Exponentials: e^{a+b} = e^a * e^b

Theorems

Euler's Identity
Properties of Complex Exponentials

Suitable Grade Level

Undergraduate (Mathematics/Engineering)