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    Fourier Series

    Uploaded by

    Dan Roger Ibale
    okok

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    © All Rights Reserved
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    Fourier Series

    Uploaded by

    Dan Roger Ibale
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    7 views7 pages

    Fourier Series

    Uploaded by

    Dan Roger Ibale
    okok

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 7

    FOURIER

    SERIES
    1.0 INTRODUCTION
    A FOURIER SERIES is an infinite trigonometric series of
    a PERIODIC FUNCTION which is continuous or have
    finite discontinuities. The fourier series of a 𝑓(𝑥) of a
    period 2L where L denotes the half period which is
    given by:
    𝒂𝟎 𝝅𝒙 𝟐𝝅𝒙 𝟑𝝅𝒙 𝝅𝒙 𝟐𝝅𝒙 𝟑𝝅𝒙
    𝒇 𝒙 = + 𝒂𝟏 𝒄𝒐𝒔 + 𝒂𝟐 𝒄𝒐𝒔 + 𝒂𝟑 𝒄𝒐𝒔 + ⋯ + 𝒃𝟏 𝒔𝒊𝒏 + 𝒃𝟐 𝒔𝒊𝒏 + 𝒃𝟑 𝒔𝒊𝒏 +⋯
    𝟐 𝑳 𝑳 𝑳 𝑳 𝑳 𝑳

    Summation Notation of Fourier Series


    ∞
    𝒂𝟎 𝝅𝒏𝒙 𝝅𝒏𝒙
    𝒇 𝒙 = + ෍ 𝒂𝒏 𝒄𝒐𝒔 + 𝒃𝒏 𝒔𝒊𝒏( )
    𝟐 𝑳 𝑳
    𝒏=𝟏

    where 𝒂𝟎 , 𝒂𝟏 , 𝒂𝟐 , 𝒂𝟑 … 𝒃𝟏 , 𝒃𝟐 , 𝒃𝟑 are known as fourier series constants

    Note: Fourier Series are only applicable for periodic and continuous functions as implied
    by its definition. In special cases where there is a discontinuity it should only be finite
    such that we can still take the fourier series of that function 𝒇 𝒙
    1.1 FINDING THE FOURIER SERIES CONSTANT
    The fourier series constants can be calculated using the general equation and some special
    cases.

    General Formula: Special Case 1:

    𝑐+2𝐿 • If 𝒇(𝒙) is an even function then 𝑏𝑛 = 0 and the


    1 fourier series coefficients are obtained as follows
    𝑎0 = න 𝑓 𝑥 𝑑𝑥
    𝐿 𝑐
    2 𝐿
    𝑎0 = න 𝑓 𝑥 𝑑𝑥
    𝐿 0
    1 𝑐+2𝐿 𝜋𝑛𝑥
    𝑎𝑛 = න 𝑓 𝑥 cos( )𝑑𝑥 2 𝐿 𝜋𝑛𝑥
    𝐿 𝑐 𝐿
    𝑎𝑛 = න 𝑓 𝑥 cos( )𝑑𝑥
    𝐿 0 𝐿
    1 𝑐+2𝐿 𝜋𝑛𝑥
    𝑏𝑛 = න 𝑓 𝑥 sin( )𝑑𝑥 Special Case 2:
    𝐿 𝑐 𝐿
    • If 𝒇(𝒙) is an odd function then 𝑎0 = 0 and 𝑎𝑛 = 0 in
    such case 𝑏𝑛 can be obtained using
    2 𝐿 𝜋𝑛𝑥
    𝑏𝑛 = න 𝑓 𝑥 sin( )𝑑𝑥
    𝐿 0 𝐿
    1.2 ODD AND EVEN FUNCTIONS
    EVEN FUNCTIONS ODD FUNCTIONS

    A function is said to be even if 𝑓 −𝑥 = 𝑓(𝑥). Graphically the graph of A function is said to be odd if 𝑓 −𝑥 = −𝑓(𝑥). Graphically the graph
    𝑓(𝑥) is said to be even if its symmetrical about the vertical y-axis. of 𝑓(𝑥) is said to be odd if its symmetrical about the origin.
    (Example: 𝑥 2 , 𝑐𝑜𝑠𝑥, sin2 𝑥) (Example: x, 𝑠𝑖𝑛𝑥, 𝑥 3 )
    1.3 TRIGONOMETRIC IDENTITIES
    Recall on Trigonometric Identities Summary on Evaluation of sine and
    Product and Sum identities cosine values

    1
    Sine Values Cosine Values
    1. 𝑠𝑖𝑛𝐴𝑐𝑜𝑠𝐵 =
    2
    sin 𝐴 − 𝐵 + sin 𝐴 + 𝐵 sin 0 = 0 cos 0 = 1
    1 𝑛
    2. 𝑐𝑜𝑠𝐴𝑐𝑜𝑠𝐵 = cos 𝐴 − 𝐵 + cos 𝐴 + 𝐵 sin 𝜋𝑛 = sin −𝜋𝑛 = 0 cos 𝜋𝑛 = cos −𝜋𝑛 = −1
    2
    1 sin 2𝜋𝑛 = 0 cos 2𝜋𝑛 = 1
    3. 𝑠𝑖𝑛𝐴𝑠𝑖𝑛𝐵 =
    2
    cos 𝐴 − 𝐵 − cos 𝐴 + 𝐵
    𝑛+1
    −sin 𝜋𝑛 = 0 −cos 𝜋𝑛 = −1

    Recall on Integration Odd and Even “n” representation

    1.
    𝜋 1 𝜋 If 𝑛 = 1,2,3,4,5,6, … , then 2𝑛 − 1 represents odd
    ‫׬‬−𝜋 sin(𝑛𝑥) 𝑑𝑥 = − cos(𝑛𝑥) −𝜋 =0
    𝑛 numbers
    𝜋 1 𝜋
    2. ‫׬‬−𝜋 cos(𝑛𝑥) 𝑑𝑥 =
    𝑛
    sin(𝑛𝑥) −𝜋 =0 If 𝑛 = 1,2,3,4,5,6, … , then 2𝑛 represents even numbers
    1.4 GRAPHS OF SOME BASIC FUNCTIONS
    1.5 EXAMPLES
    1 −𝜋<𝑥<0
    1. 𝑓 𝑥 = ቊ 0<𝑥<𝜋
    0

    a) Graph the function


    b) Determine its Fourier Series
    𝜋 1 1 1
    c) Prove that by assigning value to x will result to = 1 − + − + ⋯
    4 3 5 7

    0 −𝜋<𝑥<0
    2. 𝑓 𝑥 = ቊ 0<𝑥<𝜋
    𝑥

    a) Graph the function


    b) Determine its Fourier Series

    𝜋−𝑡 −𝜋 < 𝑥 < 0


    3. 𝑓 𝑥 = ቐ 𝜋 + 𝑡 0<𝑥<𝜋
    𝑓(𝑡 + 2𝜋)

    a) Graph the function


    b) Determine its Fourier Series

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