Math Problem Statement
answer and follow the instruction
Solution
We are tasked with finding the Fourier series representation of the given periodic function based on the provided piecewise plot, up to the 5th harmonic. I will outline the detailed steps to calculate the Fourier series using the information in the figure and proceed to compute the Fourier coefficients.
Step 1: Analyze the given function
From the graph:
- The period , which means the fundamental angular frequency is .
- The function values based on the points marked in the graph are:
- The function appears to be piecewise linear.
The function is periodic with a period , so we will calculate the Fourier series in the form: where is the fundamental angular frequency.
Step 2: Fourier coefficients formulas
We use the following formulas for the Fourier coefficients:
- is the average value over one period:
- (for ) is the cosine Fourier coefficient:
- (for ) is the sine Fourier coefficient:
Given that the function is defined piecewise linearly, we will need to break the integral calculations into segments according to the different linear pieces of the graph.
Step 3: Calculate
We begin by calculating , which is the average value of the function over one period.
Since the function is linear and symmetric over the interval , I will calculate the areas piecewise.
Step 4: Calculate and
Next, we will compute and using the piecewise segments, and then express the Fourier series up to the 5th harmonic.
Let me perform the calculations for these coefficients.### Step 5: Results of Fourier coefficients
-
The DC term (the average value of the function):
-
The cosine coefficients for the first five harmonics:
-
The sine coefficients for the first five harmonics are all zero:
Step 6: Fourier series up to the 5th harmonic
The Fourier series representation of the function , up to the 5th harmonic, is:
This series captures the function's behavior with its fundamental frequency and up to the 5th harmonic.
Would you like further details on any part of the solution or any clarification?
Questions for further exploration:
- How is the Fourier series affected if the period changes?
- Can you explain the role of harmonics in approximating periodic functions?
- What would happen if the function had discontinuities in the graph?
- How would the Fourier series change if sine terms were non-zero?
- What practical applications can this Fourier series be used for in engineering?
Tip:
For smoother functions, higher harmonics contribute less, meaning we can often approximate the function well using only a few terms in the Fourier series.
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Math Problem Analysis
Mathematical Concepts
Fourier Series
Piecewise Functions
Integral Calculus
Formulas
Fundamental frequency: omega_0 = 1
Average value over period (a_0): a_0 = 1/(2*pi) * integral from 0 to 2*pi of f(t) dt
Cosine Fourier coefficients (a_n): a_n = 2/T * integral from 0 to T of f(t) cos(n*omega_0*t) dt
Sine Fourier coefficients (b_n): b_n = 2/T * integral from 0 to T of f(t) sin(n*omega_0*t) dt
Theorems
Fourier Series Expansion
Suitable Grade Level
University Level (Undergraduate in Engineering or Mathematics)
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