Harmonics Only • Symmetry Examples • Summary E1 10 Fourier Series and Transforms (2014-5543) Complex Fourier Series: 3 – 2 / 12 Euler's Equation: e
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3: Complex Fourier Series
3: Complex Fourier Series•Euler's Equation
•Complex Fourier Series •Averaging ComplexExponentials
•Complex Fourier Analysis •Fourier Series↔Complex Fourier Series
•Complex Fourier AnalysisExample
•Time Shifting •Even/Odd Symmetry •Antiperiodic?OddHarmonics Only
•Symmetry Examples •Summary E1.10 Fourier Series and Transforms (2014-5543)Complex Fourier Series: 3 - 1 / 12Euler's Equation
3: Complex Fourier Series•Euler's Equation•Complex Fourier Series
•Averaging ComplexExponentials
•Complex Fourier Analysis •Fourier Series↔Complex Fourier Series
•Complex Fourier AnalysisExample
•Time Shifting •Even/Odd Symmetry •Antiperiodic?OddHarmonics Only
•Symmetry Examples •Summary E1.10 Fourier Series and Transforms (2014-5543)Complex Fourier Series: 3 - 2 / 12Euler's Equation:
eiθ= cosθ+isinθ [see RHB 3.3]Euler's Equation
3: Complex Fourier Series•Euler's Equation•Complex Fourier Series
•Averaging ComplexExponentials
•Complex Fourier Analysis •Fourier Series↔Complex Fourier Series
•Complex Fourier AnalysisExample
•Time Shifting •Even/Odd Symmetry •Antiperiodic?OddHarmonics Only
•Symmetry Examples •Summary E1.10 Fourier Series and Transforms (2014-5543)Complex Fourier Series: 3 - 2 / 12Euler's Equation:
eiθ= cosθ+isinθ [see RHB 3.3]Hence:
cosθ=eiθ+e-iθ 2Euler's Equation
3: Complex Fourier Series•Euler's Equation•Complex Fourier Series
•Averaging ComplexExponentials
•Complex Fourier Analysis •Fourier Series↔Complex Fourier Series
•Complex Fourier AnalysisExample
•Time Shifting •Even/Odd Symmetry •Antiperiodic?OddHarmonics Only
•Symmetry Examples •Summary E1.10 Fourier Series and Transforms (2014-5543)Complex Fourier Series: 3 - 2 / 12Euler's Equation:
eiθ= cosθ+isinθ [see RHB 3.3]Hence:
cosθ=eiθ+e-iθ 2 sinθ=eiθ-e-iθ 2iEuler's Equation
3: Complex Fourier Series•Euler's Equation•Complex Fourier Series
•Averaging ComplexExponentials
•Complex Fourier Analysis •Fourier Series↔Complex Fourier Series
•Complex Fourier AnalysisExample
•Time Shifting •Even/Odd Symmetry •Antiperiodic?OddHarmonics Only
•Symmetry Examples •Summary E1.10 Fourier Series and Transforms (2014-5543)Complex Fourier Series: 3 - 2 / 12Euler's Equation:
eiθ= cosθ+isinθ [see RHB 3.3]Hence:
cosθ=eiθ+e-iθ 2=12eiθ+1
2e-iθ
sinθ=eiθ-e-iθ 2i=-12ieiθ+1
2ie-iθ
Euler's Equation
3: Complex Fourier Series•Euler's Equation•Complex Fourier Series
•Averaging ComplexExponentials
•Complex Fourier Analysis •Fourier Series↔Complex Fourier Series
•Complex Fourier AnalysisExample
•Time Shifting •Even/Odd Symmetry •Antiperiodic?OddHarmonics Only
•Symmetry Examples •Summary E1.10 Fourier Series and Transforms (2014-5543)Complex Fourier Series: 3 - 2 / 12Euler's Equation:
eiθ= cosθ+isinθ [see RHB 3.3]Hence:
cosθ=eiθ+e-iθ 2=12eiθ+1
2e-iθ
sinθ=eiθ-e-iθ 2i=-12ieiθ+1
2ie-iθ
Most maths becomes simpler if you useeiθinstead ofcosθandsinθEuler's Equation
3: Complex Fourier Series•Euler's Equation•Complex Fourier Series
•Averaging ComplexExponentials
•Complex Fourier Analysis •Fourier Series↔Complex Fourier Series
•Complex Fourier AnalysisExample
•Time Shifting •Even/Odd Symmetry •Antiperiodic?OddHarmonics Only
•Symmetry Examples •Summary E1.10 Fourier Series and Transforms (2014-5543)Complex Fourier Series: 3 - 2 / 12Euler's Equation:
eiθ= cosθ+isinθ [see RHB 3.3]Hence:
cosθ=eiθ+e-iθ 2=12eiθ+1
2e-iθ
sinθ=eiθ-e-iθ 2i=-12ieiθ+1
2ie-iθ
Most maths becomes simpler if you useeiθinstead ofcosθandsinθExamples where usingeiθmakes things simpler:
Usingeiθ
Usingcosθandsinθ
ei(θ+φ)=eiθeiφEuler's Equation
3: Complex Fourier Series•Euler's Equation•Complex Fourier Series
•Averaging ComplexExponentials
•Complex Fourier Analysis •Fourier Series↔Complex Fourier Series
•Complex Fourier AnalysisExample
•Time Shifting •Even/Odd Symmetry •Antiperiodic?OddHarmonics Only
•Symmetry Examples •Summary E1.10 Fourier Series and Transforms (2014-5543)Complex Fourier Series: 3 - 2 / 12Euler's Equation:
eiθ= cosθ+isinθ [see RHB 3.3]Hence:
cosθ=eiθ+e-iθ 2=12eiθ+1
2e-iθ
sinθ=eiθ-e-iθ 2i=-12ieiθ+1
2ie-iθ
Most maths becomes simpler if you useeiθinstead ofcosθandsinθExamples where usingeiθmakes things simpler: