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MATH 461: Fourier Series and Boundary Value Problems ...

MATH 461: Fourier Series and Boundary ValueProblemsChapter III: Fourier SeriesGreg FasshauerDepartment of Applied MathematicsIllinois Institute of TechnologyFall 461 Chapter 31 Outline1 Piecewise Smooth functions and periodic Extensions2 Convergence of Fourier Series3 Fourier Sine and Cosine Series4 Term-by-Term Differentiation of Fourier Series5 Integration of Fourier Series6 Complex Form of Fourier 461 Chapter 32 Piecewise Smooth functions and periodic ExtensionsDefinitionA functionf, defined on[a,b], is piecewise continuous if it iscontinuous on[a,b]except at finitely many points. If bothfandf arepiecewise continuous, thenfis called piecewise means that the graphs of f and f may have only finitely manyfinite 461 Chapter 34 Piecewise Smooth functions and periodic 461 Chapter 35 ExampleThe functionf(x) =|x|defined on <x< ispiecewise smoothsincefis continuous throughout theinterval,andf is discontinuous only atx= functionf(x) ={x2, <x<0x2+1,0 x< ispiecewise smooth since bothfandf are continuous except atx= Smooth functions and periodic 461 Chapter 36 ExampleThe functionf(x) ={ ln(1 x),0 x<11,1 x<2isnot piecewise continuous (andtherefore also not piec)}}

Outline 1 Piecewise Smooth Functions and Periodic Extensions 2 Convergence of Fourier Series 3 Fourier Sine and Cosine Series 4 Term-by-Term Differentiation of Fourier Series 5 Integration of Fourier Series 6 Complex Form of Fourier Series fasshauer@iit.edu MATH 461 …

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