# Engineering Mathematics: Fourier Series Formula pdf

An online fourier series formulas printable

## 1. Fourier Series – IntroductionJean Baptiste Joseph Fourier (1768-1830) was a French mathematician, physicist and engineer, and the founder of Fourier analysis.Fourier series are used in the analysis of periodic functions. The Fourier transform and Fourier’s law are also named in his honour. |

### 2. Fourier Series of Even and Odd Functions

A function f(x) is said to be even if f(-x) = f(x).

The function f(x) is said to be odd if f(-x) = -f(x)

Graphically, even functions have symmetry about the y-axis,whereas odd functions have symmetry around the origin.

**Examples:**

Sums of odd powers of x are odd: 5x^{3}– 3x

Sums of even powers of x are even: -x^{6} + 4x^{4}+ x^{2}-3

sin x is odd, and cos x is even

The product of two odd functions is even: x sin x is even

The product of two even functions is even: x^{2}cos x is even

The product of an even function and an odd function is odd: sin x cos x is odd

### 3. Integrating even functions over symmetric domains.

Let p > 0 be any fixed number. If f(x) is an odd function, then

Let p > 0 be any fixed number. If f(x) is an even function, then

### 4. Periodic functions

__Definition:__

A function f(x) is said to be periodic if there exists a number

T > 0 such that f(x + T) = f(x) for every x. The smallest such

T is called the period of f(x).

Intuition: periodic functions have repetitive behavior.A periodic function can be defined on a finite interval,

then copied and pasted so that it repeats itself.

### 5. The fourier series of the function f(x)

a(k) =f(x) cos kx dx

b(k) =f(x) sin kx dx

**6. Remainder of fourier series**

Sn(x) = sum of first n+1 terms at x.

**remainder(n) = f(x) – Sn(x) = f(x+t) Dn(t) dt**

**Sn(x) = f(x+t) Dn(t) dt**

D_{n}(x) = Dirichlet kernel =

#### Comments

The Dirichlet kernel is also called the Dirichlet summation kernel. There is also a different normalization in use: the kernels D_{n} and

**7. Riemann’s Theorem**

.

If f(x) is continuous except for a finite # of finite jumps in every finite interval then:

**lim _{(k->)} f(t) cos kt dt = lim_{(k-> )} f(t) sin kt dt = 0**

**The fourier series of the function f(x) in an arbitrary interval.**

**A(0) / 2 + (k=1..) [ A(k) cos (k(Π)x / m) + B(k) (sin k(Π)x / m) ]**

a(k) = 1/mf(x) cos (k(Π)x / m) dx

b(k) = 1/mf(x) sin (k(Π)x / m) dx

**8. Parseval’s Theorem**

.

Parseval’s theorem usually refers to the result that the Fourier transform is unitary, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform.

If f(x) is continuous; f(-PI) = f(PI) then

^{2}(x) dx = a(0)^{2} / 2 + ^{2} + b(k)^{2})

**Fourier Integral of the function f(x)**

**f(x) = ( a(y) cos yx + b(y) sin yx ) dy**

a(y) =f(t) cos ty dt

b(y) =f(t) sin ty dt

**f(x) = dy f(t) cos (y(x-t)) dt**

**9. Special Cases of Fourier Integral**

**if f(x) = f(-x) then**

f(x) =cos xy dy f(t) cos yt dt

**if f(-x) = -f(x) then**

f(x) =sin xy dy sin yt dt

**10. The Fourier Transforms**

__Fourier Cosine Transform__

**g(x) = () f(t) cos xt dt**

__Fourier Sine Transform__

**g(x) = () f(t) sin xt dt**

**11. Identities of the Transforms**

If f(-x) = f(x) then

Fourier Cosine Transform ( Fourier Cosine Transform (f(x)) ) = f(x)

If f(-x) = -f(x) then

Fourier Sine Transform (Fourier Sine Transform (f(x)) ) = f(x)