COMPLEX FORM OF FOURIER SERIES
7. COMPLEX FORM OF FOURIER SERIES
Recall that real sines and cosines can be expressed in terms of complex exponentials by the formulas [Chapter 2, (11.3)]
e inx +e −inx (7.1)
e inx −e −inx
sin nx =
cos nx =
2 If we substitute equations (7.1) into a Fourier series like (5.12), we get a series of
2i
terms of the forms e inx and e −inx . This is the complex form of a Fourier series. We can also find the complex form directly; this is often easier than finding the sine- cosine form. We can then, if we like, work back the other way and [using Euler’s formula, Chapter 2, (9.3)] get the sine-cosine form from the exponential form.
We want to see how to find the coefficients in the complex form directly. We assume a series
f (x) = c 0 +c 1 e ix +c −1 e −ix +c 2 e 2ix +c −2 e −2ix +···
n=+∞
c n e inx
n=−∞
Section 7 Complex Form of Fourier Series 359 and try to find the c n ’s. From (5.4) we know that the average value of e ikx on (−π, π)
is zero when k is an integer not equal to zero. To find c 0 , we find the average values of the terms in (7.2):
1 π 1 π average values of terms of the (7.3)
form e −π ikx −π
To find c n , we multiply (7.2) by e −inx and again find the average value of each term. Note the minus sign in the exponent. In finding a n , the coefficient of cos nx in equation (5.1), we multiplied by cos nx; but here in finding the coefficient c n of
e inx , we multiply by the complex conjugate e −inx .
dx = c 0 e −inx dx + c 1 e −inx e ix dx 2π −π
f (x)e −inx
e −inx e −ix dx + · · · .
The terms on the right are the average values of exponentials e ikx , where the k values are integers. Therefore all these terms are zero except the one where k = 0; this is the term containing c n . We then have
dx = c n · dx = c n , 2π −π
f (x)e −inx dx = c
e −inx e inx
f (x)e −inx dx.
Note that this formula contains the one for c 0 (no 1 2 to worry about here!). Also, since (7.6) is valid for negative as well as positive n, you have only one formula to memorize here! You can easily show that for real f (x), c −n =¯ c n (Problem 12).
Example. Let us expand the same f (x) we did before, namely (5.11). We have from (7.6)
e −inx · 0 · dx +
e −inx · 1 · dx
(7.7) = 1 = πin (e , n odd, −inπ − 1) =
1 e −inx π
2π −in 0 −2πin
dx = .
360 Fourier Series and Transforms Chapter 7
Then
e 5ix (7.8)
e −3ix
e −5ix
It is interesting to verify that this is the same as the sine-cosine series we had before. We could use Euler’s formula for each exponential, but it is easier to collect terms like this:
which is the same as (5.12).
PROBLEMS, SECTION 7
1 to 11. Expand the same functions as in Problems 5.1 to 5.11 in Fourier series of complex exponentials e inx on the interval (−π, π) and verify in each case that the answer is equivalent to the one found in Section 5.
12. Show that if a real f (x) is expanded in a complex exponential Fourier series P ∞
−∞ c n e inx , then c −n =¯ c n , where ¯ c n means the complex conjugate of c n .
−∞ c n e , use Euler’s formula to find a n and b n in terms of c n and c −n , and to find c n and c −n in terms of a n
Parts
» CONVERGENCE TESTS FOR SERIES OF POSITIVE TERMS; ABSOLUTE CONVERGENCE
» CONDITIONALLY CONVERGENT SERIES
» POWER SERIES; INTERVAL OF CONVERGENCE
» EXPANDING FUNCTIONS IN POWER SERIES
» TECHNIQUES FOR OBTAINING POWER SERIES EXPANSIONS
» ACCURACY OF SERIES APPROXIMATIONS
» COMPLEX POWER SERIES; DISK OF CONVERGENCE
» ELEMENTARY FUNCTIONS OF COMPLEX NUMBERS
» POWERS AND ROOTS OF COMPLEX NUMBERS
» THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
» LINEAR COMBINATIONS, LINEAR FUNCTIONS, LINEAR OPERATORS
» LINEAR DEPENDENCE AND INDEPENDENCE
» SPECIAL MATRICES AND FORMULAS
» EIGENVALUES AND EIGENVECTORS; DIAGONALIZING MATRICES
» APPLICATIONS OF DIAGONALIZATION
» A BRIEF INTRODUCTION TO GROUPS
» POWER SERIES IN TWO VARIABLES
» APPROXIMATIONS USING DIFFERENTIALS
» CHAIN RULE OR DIFFERENTIATING A FUNCTION OF A FUNCTION
» APPLICATION OF PARTIAL DIFFERENTIATION TO MAXIMUM AND MINIMUM PROBLEMS
» MAXIMUM AND MINIMUM PROBLEMS WITH CONSTRAINTS; LAGRANGE MULTIPLIERS
» ENDPOINT OR BOUNDARY POINT PROBLEMS
» DIFFERENTIATION OF INTEGRALS; LEIBNIZ’ RULE
» APPLICATIONS OF INTEGRATION; SINGLE AND MULTIPLE INTEGRALS
» CHANGE OF VARIABLES IN INTEGRALS; JACOBIANS
» APPLICATIONS OF VECTOR MULTIPLICATION
» DIRECTIONAL DERIVATIVE; GRADIENT
» SOME OTHER EXPRESSIONS INVOLVING ∇
» GREEN’S THEOREM IN THE PLANE
» THE DIVERGENCE AND THE DIVERGENCE THEOREM
» THE CURL AND STOKES’ THEOREM
» SIMPLE HARMONIC MOTION AND WAVE MOTION; PERIODIC FUNCTIONS
» APPLICATIONS OF FOURIER SERIES
» COMPLEX FORM OF FOURIER SERIES
» LINEAR FIRST-ORDER EQUATIONS
» OTHER METHODS FOR FIRST-ORDER EQUATIONS
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND ZERO RIGHT-HAND SIDE
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND RIGHT-HAND SIDE NOT ZERO
» OTHER SECOND-ORDER EQUATIONS
» SOLUTION OF DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS
» A BRIEF INTRODUCTION TO GREEN FUNCTIONS
» THE BRACHISTOCHRONE PROBLEM; CYCLOIDS
» SEVERAL DEPENDENT VARIABLES; LAGRANGE’S EQUATIONS
» TENSOR NOTATION AND OPERATIONS
» KRONECKER DELTA AND LEVI-CIVITA SYMBOL
» PSEUDOVECTORS AND PSEUDOTENSORS
» VECTOR OPERATORS IN ORTHOGONAL CURVILINEAR COORDINATES
» ELLIPTIC INTEGRALS AND FUNCTIONS
» GENERATING FUNCTION FOR LEGENDRE POLYNOMIALS
» COMPLETE SETS OF ORTHOGONAL FUNCTIONS
» NORMALIZATION OF THE LEGENDRE POLYNOMIALS
» THE ASSOCIATED LEGENDRE FUNCTIONS
» GENERALIZED POWER SERIES OR THE METHOD OF FROBENIUS
» THE SECOND SOLUTION OF BESSEL’S EQUATION
» OTHER KINDS OF BESSEL FUNCTIONS
» ORTHOGONALITY OF BESSEL FUNCTIONS
» SERIES SOLUTIONS; FUCHS’S THEOREM
» HERMITE FUNCTIONS; LAGUERRE FUNCTIONS; LADDER OPERATORS
» LAPLACE’S EQUATION; STEADY-STATE TEMPERATURE IN A RECTANGULAR PLATE
» THE DIFFUSION OR HEAT FLOW EQUATION; THE SCHR ¨ODINGER EQUATION
» THE WAVE EQUATION; THE VIBRATING STRING
» STEADY-STATE TEMPERATURE IN A CYLINDER
» VIBRATION OF A CIRCULAR MEMBRANE
» STEADY-STATE TEMPERATURE IN A SPHERE
» INTEGRAL TRANSFORM SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
» EVALUATION OF DEFINITE INTEGRALS BY USE OF THE RESIDUE THEOREM
» THE POINT AT INFINITY; RESIDUES AT INFINITY
» SOME APPLICATIONS OF CONFORMAL MAPPING
» THE NORMAL OR GAUSSIAN DISTRIBUTION
» STATISTICS AND EXPERIMENTAL MEASUREMENTS
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