Fourier Series

Fourier Analysis Discrete Continuous Fourier Series Fourier Integral Fast Fourier Transform Discrete Fourier Transform Fourier Transform

Outline Periodic Function Fourier Cosine and Sine Series Periodic Function with Period 2L Odd and Even Functions Half Range Fourier Cosine and Sine Series Complex Notation for Fourier Series

Fourier, Joseph Fourier, Joseph 1768-1830

Fourier, Joseph In 1807, Fourier submitted a paper to the Academy of Sciences of Paris. In it he derived the heat equation and proposed his separation of variables method of solution. The paper, evaluated by Laplace, Lagrange, and Lagendre, was rejected for lack rigor. However, the results were promising enough for the academy to include the problem of describing heat conduction in a prize competition in 1812. Fourier’s 1811 revision of his earlier paper won the prize, but suffered the same criticism as before. In 1822, Fourier finally published his classic Theorie analytique de la chaleur, laying the fundations not only for the separation of variables method and Fourier series, but for the Fourier integral and transform as well.

Periodic Function Definition: Periodic Function A function f(x) is said to be periodic with period T if for all x f(x) x T

Periodic Function f(x+p)=f(x), f(x+np)=f(x) If f(x) and g(x) have period p, the the function H(x)=af(x)+bg(x) , also has the period p If a period function of f(x) has a smallest period p (p >0), this is often called the fundamental period of f(x)

Periodic Function Example Cosine Functions: cosx, cos2x, cos3x, … Sine Functions: sinx, sin2x, sin3x, … eix, ei2x, ei3x, … e-ix, e-i2x, e-i3x, …

Fourier Cosine and Sine Series Lemma: Trigonometric System is Orthogonal

Fourier Cosine and Sine Series A function f(x) is periodic with period 2 and

Fourier Cosine and Sine Series (Euler formulas) Then

Fourier Cosine and Sine Series Proof:

Fourier Cosine and Sine Series Example 5-1: Find the Fourier coefficients corresponding to the function Sol:

Fourier Cosine and Sine Series Sol:

Fourier Cosine and Sine Series Sol:

Periodic Function with Period 2L A periodic function f(x) with period 2L f(x) x 2L

Periodic Function with Period 2L Then

Periodic Square Wave

Odd and Even Functions A function f(x) is said to be even if A function f(x) is said to be odd if

Odd and Even Functions Even Function Odd Function f(x) f(x) x x

Odd and Even Functions Property The product of an even and an odd function is odd.

Odd and Even Functions Fourier Cosine Series Fourier Sine Series

Sun of Functions The Fourier coefficients of a sum f1+f2 are the sum of the corresponding Fourier coefficients of f1 and f2. The Fourier coefficients of cf are c times the corresponding Fourier coefficients of f.

Examples Rectangular Pulse Sawtooth wave The function f*(x) is the sum in Example 1 of Sec.10.2 and the constant k. Sawtooth wave Find the Fourier series of the function

Half-Range Expansions A function f is given only on half the range, half the interval of periodicity of length 2L. even periodic extention f1 of f odd period extention f2 of f

Complex Notation for Fourier Series

Complex Notation for Fourier Series A periodic function f(x) with period 2L

Complex Fourier Series Find complex Fourier series

Exercise Section 10-4 Section 10-2 Section 10-3 #1, #5, #11 #5, #9 #1, #15

Fourier Cosine and Sine Integrals Example 1

Fourier Cosine and Sine Integrals sin(x) x

Fourier Cosine and Sine Integrals x

Fourier Cosine and Sine Integrals

Fourier Cosine and Sine Integrals

Fourier Cosine and Sine Integrals

Gibb’s Phenomenon Sine Integral

Gibb’s Phenomenon Gibb’s Phenomenon

Gibb’s Phenomenon

Fourier Cosine and Sine Integrals Fourier Cosine Integral of f(t)

Fourier Cosine and Sine Integrals Fourier Sine Integral of f(t)

Fourier Integrals Fourier Integral of f(t)