- Source: Fourier sine and cosine series
In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after Joseph Fourier.
Notation
In this article, f denotes a real-valued function on
R
{\displaystyle \mathbb {R} }
which is periodic with period 2L.
Sine series
If f is an odd function with period
2
L
{\displaystyle 2L}
, then the Fourier Half Range sine series of f is defined to be
f
(
x
)
=
∑
n
=
1
∞
b
n
sin
(
n
π
x
L
)
{\displaystyle f(x)=\sum _{n=1}^{\infty }b_{n}\sin \left({\frac {n\pi x}{L}}\right)}
which is just a form of complete Fourier series with the only difference that
a
0
{\displaystyle a_{0}}
and
a
n
{\displaystyle a_{n}}
are zero, and the series is defined for half of the interval.
In the formula we have
b
n
=
2
L
∫
0
L
f
(
x
)
sin
(
n
π
x
L
)
d
x
,
n
∈
N
.
{\displaystyle b_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {n\pi x}{L}}\right)\,dx,\quad n\in \mathbb {N} .}
Cosine series
If f is an even function with a period
2
L
{\displaystyle 2L}
, then the Fourier cosine series is defined to be
f
(
x
)
=
a
0
2
+
∑
n
=
1
∞
a
n
cos
(
n
π
x
L
)
{\displaystyle f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }a_{n}\cos \left({\frac {n\pi x}{L}}\right)}
where
a
n
=
2
L
∫
0
L
f
(
x
)
cos
(
n
π
x
L
)
d
x
,
n
∈
N
0
.
{\displaystyle a_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\cos \left({\frac {n\pi x}{L}}\right)\,dx,\quad n\in \mathbb {N} _{0}.}
Remarks
This notion can be generalized to functions which are not even or odd, but then the above formulas will look different.
See also
Fourier series
Fourier analysis
Least-squares spectral analysis
Bibliography
Byerly, William Elwood (1893). "Chapter 2: Development in Trigonometric Series". An Elementary Treatise on Fourier's Series: And Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics (2 ed.). Ginn. p. 30.
Carslaw, Horatio Scott (1921). "Chapter 7: Fourier's Series". Introduction to the Theory of Fourier's Series and Integrals, Volume 1 (2 ed.). Macmillan and Company. p. 196.
Kata Kunci Pencarian:
- Sinus dan kosinus
- Fourier sine and cosine series
- Fourier series
- Sine and cosine
- Fourier analysis
- Fourier transform
- List of things named after Joseph Fourier
- List of Fourier-related transforms
- Discrete sine transform
- Trigonometric integral
- Sine wave
