Odd 3: Complex Fourier Series

1 3: Complex Fourier Series3: 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 1 / 12 Euler s Equation3: 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 2 / 12 Euler s Equation:ei = cos +isin [see RHB ]Euler s Equation3.

2 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 2 / 12 Euler s Equation:ei = cos +isin [see RHB ]Hence:cos =ei +e i 2 Euler s Equation3: 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 2 / 12 Euler s Equation:ei = cos +isin [see RHB ]Hence:cos =ei +e i 2sin =ei e i 2iEuler s Equation3.

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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 2 / 12 Euler s Equation:ei = cos +isin [see RHB ]Hence:cos =ei +e i 2=12ei +12e i sin =ei e i 2i= 12iei +12ie i Euler s Equation3: 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 2 / 12 Euler s Equation:ei = cos +isin [see RHB ]Hence:cos =ei +e i 2=12ei +12e i sin =ei e i 2i= 12iei +12ie i Most maths becomes simpler if you useei instead ofcos andsin Euler s Equation3.

4 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 2 / 12 Euler s Equation:ei = cos +isin [see RHB ]Hence:cos =ei +e i 2=12ei +12e i sin =ei e i 2i= 12iei +12ie i Most maths becomes simpler if you useei instead ofcos andsin Examples where usingei makes things simpler:Usingei Usingcos andsin ei( + )=ei ei Euler s Equation3: 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 2 / 12 Euler s Equation:ei = cos +isin [see RHB ]Hence:cos =ei +e i 2=12ei +12e i sin =ei e i 2i= 12iei +12ie i Most maths becomes simpler if you useei instead ofcos andsin Examples where usingei makes things simpler:Usingei Usingcos andsin ei( + )=ei ei ei ei =ei( + )Euler s Equation3.

5 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 2 / 12 Euler s Equation:ei = cos +isin [see RHB ]Hence:cos =ei +e i 2=12ei +12e i sin =ei e i 2i= 12iei +12ie i Most maths becomes simpler if you useei instead ofcos andsin Examples where usingei makes things simpler:Usingei Usingcos andsin ei( + )=ei ei ei ei =ei( + )dd ei =iei Euler s Equation3: 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 2 / 12 Euler s Equation:ei = cos +isin [see RHB ]Hence:cos =ei +e i 2=12ei +12e i sin =ei e i 2i= 12iei +12ie i Most maths becomes simpler if you useei instead ofcos andsin Examples where usingei makes things simpler:Usingei Usingcos andsin ei( + )=ei ei cos ( + ) = cos cos sin sin ei ei =ei( + )dd ei =iei Euler s Equation3.

6 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 2 / 12 Euler s Equation:ei = cos +isin [see RHB ]Hence:cos =ei +e i 2=12ei +12e i sin =ei e i 2i= 12iei +12ie i Most maths becomes simpler if you useei instead ofcos andsin Examples where usingei makes things simpler:Usingei Usingcos andsin ei( + )=ei ei cos ( + ) = cos cos sin sin ei ei =ei( + )cos cos =12cos ( + ) +12cos ( )dd ei =iei Euler s Equation3: 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 2 / 12 Euler s Equation:ei = cos +isin [see RHB ]Hence.

7 Cos =ei +e i 2=12ei +12e i sin =ei e i 2i= 12iei +12ie i Most maths becomes simpler if you useei instead ofcos andsin Examples where usingei makes things simpler:Usingei Usingcos andsin ei( + )=ei ei cos ( + ) = cos cos sin sin ei ei =ei( + )cos cos =12cos ( + ) +12cos ( )dd ei =iei dd cos = sin Euler s Equation3: 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 2 / 12 Euler s Equation:ei = cos +isin [see RHB ]Hence:cos =ei +e i 2=12ei +12e i sin =ei e i 2i= 12iei +12ie i Most maths becomes simpler if you useei instead ofcos andsin TheComplex Fourier Seriesis the Fourier Series but written usingei Examples where usingei makes things simpler:Usingei Usingcos andsin ei( + )=ei ei cos ( + ) = cos cos sin sin ei ei =ei( + )cos cos =12cos ( + ) +12cos ( )dd ei =iei dd cos = sin Complex Fourier Series3.

8 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 3 / 12 Fourier Series :u(t) =a02+P n=1(ancos 2 nF t+bnsin 2 nF t) Complex Fourier Series3: 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 3 / 12 Fourier Series :u(t) =a02+P n=1(ancos 2 nF t+bnsin 2 nF t)Substitute:cos =12ei +12e i Complex Fourier Series3.

9 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 3 / 12 Fourier Series :u(t) =a02+P n=1(ancos 2 nF t+bnsin 2 nF t)Substitute:cos =12ei +12e i andsin = 12iei +12ie i Complex Fourier Series3: 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 3 / 12 Fourier Series :u(t) =a02+P n=1(ancos 2 nF t+bnsin 2 nF t)Substitute:cos =12ei +12e i andsin = 12iei +12ie i u(t) =a02+P n=1 an 12ei +12e i +bn 12iei +12ie i [ = 2 nF t] Complex Fourier Series3.

10 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 3 / 12 Fourier Series :u(t) =a02+P n=1(ancos 2 nF t+bnsin 2 nF t)Substitute:cos =12ei +12e i andsin = 12iei +12ie i u(t) =a02+P n=1 an 12ei +12e i +bn 12iei +12ie i =a02+P n=1 12an 12ibn ei2 nF t [ = 2 nF t]+P n=1 12an+12ibn e i2 nF t Complex Fourier Series3: 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 Fourier Series and Transforms (2014-5543) Complex Fourier Series : 3 3 / 12 Fourier Series :u(t) =a02+P n=1(ancos 2 nF t+bnsin 2 nF t)Substitute:cos =12ei +12e i a