find the fourier series for the piecewise n(t) = 2t+1 for 0 <= t <1, 3e^(-3(t-1)) 1 <= t <2
Question
Find the Fourier series for the piecewise
n(t) = 2t + 1 for 0 <= t < 1,
3e^{-3(t-1)} for 1 <= t < 2.
Solution
1. Break Down the Problem
To find the Fourier series of the piecewise function , we will:
- Define the function explicitly over the intervals.
- Determine the period of the function.
- Calculate the Fourier coefficients.
2. Relevant Concepts
The function is defined as:
The period of the function seems to be .
The Fourier coefficients and are calculated using the formulas:
3. Analysis and Detail
-
Calculate :
- First integral:
- Second integral (using substitution ):
- Now combine:
-
Calculate for :
- First integral (by parts or direct):
- Second integral:
-
Calculate for :
- First integral:
- Second integral:
4. Verify and Summarize
The calculations for and are more complex and may involve integration by parts or numerical methods for specific .
Final Answer
The Fourier series representation of can be given as: Where:
- and require further computation based on the integrals detailed above.
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