Lecture 3.5: Complex inner products and Fourier series Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics M. Macauley (Clemson) Lecture 3.5: Complex inner products & Fourier series Advanced Engineering Mathematics 1 / 8
Review of complex numbers Euler’s formula e i θ = cos θ + i sin θ e − i θ = cos θ − i sin θ cos θ = e i θ + e − i θ 2 sin θ = e i θ − e − i θ 2 i M. Macauley (Clemson) Lecture 3.5: Complex inner products & Fourier series Advanced Engineering Mathematics 2 / 8
Real vs. complex vector spaces Until now, we have primarily been dealing with R -vector spaces. Things are a little different with C -vector spaces. To understand why, compare the notion of norm for real vs. complex numbers. √ x 2 ∈ R . For any real number x ∈ R , its norm (distance from 0 ) is | x | = For any complex number z = a + bi ∈ C , its norm (distance from 0 ) is defined by √ � a 2 + b 2 . � | z | := zz = ( a + bi )( a − bi ) = Let’s now go from R and C to R 2 and C 2 . � v 1 � ∈ R 2 , its norm (distance from 0 ) is For any vector v = v 2 || v || = √ v · v = � � v T v = v 2 1 + v 2 2 . � z 1 � ∈ C 2 , with z 1 = a + bi , z 2 = c + di , its norm is defined by For any z = z 2 � � z T z = | z 1 | 2 + | z 2 | 2 . || z || := � 1 � � i � ∈ R 2 and v = ∈ C 2 . For example, let’s compute the norms of v = 1 i M. Macauley (Clemson) Lecture 3.5: Complex inner products & Fourier series Advanced Engineering Mathematics 3 / 8
Real vs. complex inner products Big idea The norm in an R -vector space is defined using a real inner product, e.g., v 1 . v · w := w T v = � � w 1 � · · · . = w n v i w i . . v n The norm in a C -vector space is defined using a complex inner product, e.g., z 1 . z · w := w T z = � � w 1 � · · · w n . = z i w i . . z n Definition Let V be an C -vector space. A function �− , −� : V × V → C is a (complex) inner product if it satisfies (for all u , v , w , ∈ V , c ∈ C ): (i) � u + v , w � = � u , v � + � v , w � (ii) � c v , w � = c � v , w � and � v , c w � = c � v , w � (iii) � v , w � = � w , v � (iv) � v , v � ≥ 0, with equaility if and only if v = 0. M. Macauley (Clemson) Lecture 3.5: Complex inner products & Fourier series Advanced Engineering Mathematics 4 / 8
Periodic functions as a C -vector space Recall the R -vector space (we’ll abuse terminology and allow infinite sums) � � Per 2 π ( R ) := Span { 1 , cos x , cos 2 x , . . . } ∪ { sin x , sin 2 x , . . . } Let Per 2 π ( C ) denote the same vector space but with coefficients from C . Since cos nx = e inx + e − inx sin nx = e inx − e − inx , , 2 2 i a better way to write Per 2 π ( C ) is using a different basis: . . . , e − 2 ix , e − ix , 1 , e ix , e 2 ix , . . . � � Per 2 π ( C ) := Span It turns out that this basis is orthonormal with respect to the inner product � π � f , g � := 1 f ( x ) g ( x ) dx . 2 π − π It is quite easy to verify this: � π � = 1 1 m = n e inx e imx dx = � e inx , e imx � 2 π 0 m � = n − π M. Macauley (Clemson) Lecture 3.5: Complex inner products & Fourier series Advanced Engineering Mathematics 5 / 8
Formulas for the Fourier coefficients Definition / Theorem If f ( x ) is a piecewise continuous 2 L -periodic function, then its complex Fourier series is ∞ ∞ i π nx i π nx + c − n e − i π nx � � � L � f ( x ) = c n e = c 0 + c n e L L n = −∞ n =1 where the complex Fourier coefficients are � L � L = 1 = 1 � i π nx � f ( x ) e − i π nx c 0 = � f , 1 � f ( x ) dx , c n = f , e dx . L L 2 L 2 L − L − L M. Macauley (Clemson) Lecture 3.5: Complex inner products & Fourier series Advanced Engineering Mathematics 6 / 8
Computations Example 1: square wave � 1 , 0 < x < π Find the complex Fourier series of f ( x ) = − 1 , π < x < 2 π. M. Macauley (Clemson) Lecture 3.5: Complex inner products & Fourier series Advanced Engineering Mathematics 7 / 8
Computations Example 2 Compute the complex Fourier series of the 2 π -periodic extension of the function e x defined on − π < 0 < π . M. Macauley (Clemson) Lecture 3.5: Complex inner products & Fourier series Advanced Engineering Mathematics 8 / 8
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