[PDF] [PDF] Lecture 14: Half Range Fourier Series: even and odd - UBC Math

[PDF] Sine and Cosine Series

Since f(x) is an odd function, it has a sine series expansion bn 2 ; 0 1 xsin(n=x) dx "2 n= 

[PDF] Fourier Series Expansion

f(x) cos nx dx bn = 1 π ∫ c+2π c f(x) sin nx dx 3 Derivation of Fourier series expansion of a function defined in an arbitrary period [a, b]: Now suppose that f(x ) is 

[PDF] Fourier Cosine Series Examples - MIT Mathematics

7 jan 2011 · sum of cosines, the Fourier cosine series For a function f(x) The resulting cosine-series expansion is plotted in figure 1, truncated to 1, 2, 3, 

[PDF] Lecture 14: Half Range Fourier Series: even and odd - UBC Math

(Compiled 4 August 2017) In this lecture we consider the Fourier Expansions for Even and Odd functions, which give rise to cosine and sine half range Fourier 

[PDF] 1 (a) Show that the Fourier cosine series expansion for cosax on [0

(a) Show that the Fourier cosine series expansion for cosax on [0,π] is given by cosax = 2asin aπ π [ 1 2a2 − cosx a2 − 12 + cos 2x a2 − 22 − ···] an = (−)n

[PDF] on Fourier Series - Trinity University

Transforming Fourier Series Half-range Expansions This yields −1 + 2 π ( π 2 − 4 π ∞ ∑ k=0 cos((2k + 1)π(x − 1)/2) (2k + 1)2 ) The cosine term inside 

[PDF] Commonly Used Taylor Series

Commonly Used Taylor Series series when is valid/true 1 1 − x = 1 + x + x2 + x3 + taylor seris of y = cos x has only If the power/Taylor series in formula (1)

[PDF] Taylor series expansion of f=cos(x) - UF MAE

So Taylor series expansion is (as given in Problem 4 10) 2 4 6 8 cos( ) 1 2 4 6 8 Calculates the Maclaurin series approximaton to cos(x) using the first n

[PDF] 104 Fourier Cosine and Sine Series - Berkeley Math

(0,L) This extension is called the odd 2L-periodic extension of f(x) The resulting Fourier series expansion is called a half-range expansion for f(x) because it 

[PDF] Fractional range sine and cosine series - SYCON

(HRS) series, half range cosine (HRC) series, quarter range sine (QRS) series, and quarter range cosine (QRC) series—the expansion given for a function f(x) 

[PDF] cosinus and sinus

[PDF] cosinus definition

[PDF] cosinus formule

[PDF] cosinus joint

[PDF] cosinus sinus tangens

[PDF] cosinusoidal function

[PDF] cosmetology in ancient egypt

[PDF] cost of healthcare in europe

[PDF] cost of nato by country

[PDF] cost of paris metro pass for 6 zones

[PDF] cost to cancel flight air canada

[PDF] cos^2x sin^2x=cos2x

[PDF] cotation viande bovine franceagrimer

[PDF] cotation viande franceagrimer

[PDF] cotton wilkinson basic inorganic chemistry pdf

Introductory lecture notes on Partial Differential Equations - c⃝Anthony Peirce. Not to be copied, used, or revised without explicit written permission from the copyright owner.1 Lecture 14: Half Range Fourier Series: even and odd functions (Compiled 4 August 2017)

In this lecture we consider the Fourier Expansions for Even and Odd functions, which give rise to cosine and sine half

range Fourier Expansions. If we are only given values of a functionf(x) over half of the range [0;L], we can dene two

different extensions offto the full range [L;L], which yield distinct Fourier Expansions. The even extension gives rise

to a half range cosine series, while the odd extension gives rise to a half range sine series. Key Concepts:Even and Odd Functions; Half Range Fourier Expansions; Even and Odd Extensions

14.1 Even and Odd Functions

Even:f(x) =f(x)

Odd:f(x) =f(x)

14.1.1Integrals of Even and Odd Functions

L

Lf(x)dx=0

Lf(x)dx+L

0 f(x)dx (14.1) L 0[ f(x) +f(x)]dx (14.2) 8 :2L∫

0f(x)dx feven

0fodd:

(14.3) Notes: LetE(x) represent an even function andO(x) an odd function. (1) Iff(x) =E(x)O(x) thenf(x) =E(x)O(x) =E(x)O(x) =f(x))fis odd. (2) E

1(x)E2(x)!even.

(3) O

1(x)O2(x)!even.

(4) Any function can be expressed as a sum of an even part and an odd part: f(x) =1 2 [f(x) +f(x)] {z even part+ 1 2 [f(x)f(x)] {z odd part: (14.4) 2

Check: LetE(x) =1

2 [f(x) +f(x)]. ThenE(x) =1 2 [f(x) +f(x)]=E(x) even. Similarly let

O(x) =1

2 [f(x)f(x)] (14.5)

O(x) =1

2 [f(x)f(x)]=O(x) odd: (14.6)

14.2 Consequences of the Even/Odd Property for Fourier Series

(I) Letf(x) be Even-Cosine Series: a n=1 L L

Lf(x)cos|

{z even( nx L dx=2 L L 0 f(x)cos(nx L dx (14.7) b n=1 L L

Lf(x)sin(nx

L {z odddx= 0: (14.8)

Therefore

f(x) =a0 2 +1∑ n=1a ncos(nx L ;an=2 L L 0 f(x)cos(nx L dx: (14.9) (II) Letf(x) be Odd-Sine Series: a n=1 L L

Lf(x)cos(nx

L {z odddx= 0 (14.10) b n=1 L L

Lf(x)sin(nx

L {z evendx=2 L L 0 f(x)sin(nx L dx

Therefore

f(x) =1∑ n=1b nsin(nx L ;bn=2 L L

0f(x)sin(nx

L dx:

(III) Since any function can be written as the sum of an even and odd part, we can interpret the cos and sin series

as even/odd: f(x) =evenodd 1 2 [f(x) +f(x)]+1 2 [f(x)f(x)] (14.11) a 0 2 +1∑ n=1a ncos(nx L

1∑

n=1b nsin(nx L

Fourier Series3

where a n=2 L L 01 2 [f(x) +f(x)]cos(nx L dx=1 L L

Lf(x)cos(nx

L dx b n=2 L L 01 2 [f(x)f(x)]sin(nx L dx=1 L L

Lf(x)sin(nx

L dx:

14.3 Half-Range Expansions

If we are given a functionf(x) on an interval [0;L] and we want to representfby a Fourier Series we have two

choices - a Cosine Series or a Sine Series.

Cosine Series:

f(x) =a0 2 +1∑ n=1aquotesdbs_dbs20.pdfusesText_26