- #1
jaejoon89
- 187
- 0
I'm trying to find a Fourier series for exp(-ax) where a is a positive constant. How is exp(-ax) piecewise smooth?
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SUMMARY
The function exp(-ax), where 'a' is a positive constant, is classified as piecewise smooth due to its smoothness across its entire domain. Specifically, it meets the criteria for piecewise smooth functions, as it is smooth in all but finitely many points, with the number of points of non-smoothness being zero. This characteristic allows for the derivation of a Fourier series representation for exp(-ax).
PREREQUISITES- Understanding of piecewise smooth functions
- Familiarity with Fourier series concepts
- Basic knowledge of exponential functions
- Mathematical analysis skills
- Research the properties of piecewise smooth functions
- Learn about Fourier series and their applications
- Study the behavior of exponential functions in mathematical analysis
- Explore the implications of smoothness in function theory
Mathematicians, students of advanced calculus, and anyone interested in the analysis of functions and Fourier series.
- #2
CompuChip
Science Advisor
Homework Helper
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It is piecewise smooth in one big piece 
If you take piecewise smooth to mean: smooth in all but finitely many points, it satisfies the definition because the number of points in which it is not smooth is zero ([itex]< \infty[/itex]).
If you take piecewise smooth to mean: smooth in all but finitely many points, it satisfies the definition because the number of points in which it is not smooth is zero ([itex]< \infty[/itex]).
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