J. Comp. & Math. Sci. Vol. 1 (6), 766-768 (2010)
Fourier Series of log (sin x) in (-L, L) where 0<L<π π RENE RANJAN WILLIAM, N. Ch. S. N. IYENGAR and *CHANDRA SEKHARA REDDY C. School of Computing Science and Engineering, * School of Advanced Sciences VIT University, Vellore -632014,T.N,India nchsniyengar48@gmail.com ,csreddy@vit.ac.in ABSTRACT In this paper we obtain the Fourier Series of log (sin x) in (-L, L), 0<L<π. The properties of the Fourier coefficients so obtained are also highlighted. = log(|sinx|+iβ(x))
INTRODUCTION log(sin x) is a function of special interest since real values of x can generate complex values in log(sin x). For a given value of x, there can be more than one value for log(sin x) other than it’s principle value. The graph of log(sin x) requires four dimensions. The Fourier Series of log(sin x) in (-L, L), 0<L< π is the representation of the function in a periodic form with a period length of 2L. The real and imaginary parts of the function are periodic with the same period and their Fourier Series can be found separately. Preliminaries:
∑ n =0
β nt n n!
=
0<x<L
m∈N
We derive a Fourier Series for the real and imaginary parts separately. First we consider the real part log(| sin x |) . Let the Fourier Series be of the form
f ( x) =
a0 ∞ nπ x nπ x + ∑an cos + bn sin 2 n=1 L L
Let a0 = p0+iq0, an = pn+iqn, bn = rn+isn, where po, pn, rn are the Fourier coefficients of log(| sin x |) in (-L, L), 0<L< π and qo ,qn. sn are the Fourier coefficients of β(x) in (-L, L), 0<L< π.
t . The series representation of e −1
1 P0 = ∫ log(| sin x |)dx . L −L since log(| sin x |) is an even function, L
P0 =
t
cot(x) which is convergent in |x| < π, can be expressed with the help of the Bernoulli numbers. cot( x ) =
−L < x < 0
L
Bernoulli numbers Bn can be obtained with the help of the generating function ∞
( 2 m − 1 )π where β ( x ) = 0
( −1 )n 2 2 n B2 n x 2 n −1 . ( 2n )! n =0 ∞
=
L 2 L x log(sin x )|0 − ∫ x cot( x )dx L 0
=
L ∞ ( −1 )n 22 n B2 n x 2 n −1 2 L log(sin L ) − ( dx ∫0 ∑ L ( 2n )! n=0
∑
Solution:
log(sin x ) = log(| sin x | eiβ x )
2 log(sin x )dx L ∫0
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-765)
Rene Ranjan William et al., J. Comp. & Math. Sci. Vol. 1 (6), 766-768 (2010)
767
∞ ( −1)n 22n B2n x2n−1 2 ∴P0 = Llog(sinL)−∑( L ( 2n)! n=0
Pn =
nπ x dx L
L
1 L
∫ log(| sin x |) cos
−L
=−
2 nπ x log(sin x )cos dx ∫ L0 L L
=
λ 2λ 2λ −1 2 nπ x 1 ∞ ( −1) 2 B2λ x sin + ( dx ∑ nπ ∫0 L x λ=1 ( 2λ )! L
=−
L ∞ ( −1 )λ 22 λ B2 λ L 2 λ −1 2 1 nπ x nπ x ∫ sin dx + ∑ ( ∫0 x sin L dx nπ 0 x ( 2λ )! L λ =1
The first integral is evaluated by expanding the sine series and performing term wise L L integration. The second integral is evaluated 2 nπ x nπ x L = ∫ sin log(sin x )|0 −∫ sin cot( x )dx nπ 0 L L 0 using Bernoulli integration. ∞ ( −1 )λ ( 2L )2λ B2λ 2 ∞ ( −1 )λ −1( nπ )2λ −1 n ∴ Pn = − ∑( + ( −1 ) ∑( nπ λ =1 ( 2λ −1 )( 2λ −1 )! 2λ λ =1 nπ x rn = 0 since log(| sin x |) sin is an odd function. L
( −1 )k ( ∑ 2 k −1 ( 2λ − 2k + 1 )! k =1 ( nπ ) λ
Now, we proceed to finding the Fourier Series of the imaginary part β(x). L 1 0 1 q0 = ∫ β( x)dx = ∫ ( 2m −1)π dx = ( 2m −1)π L −L L −L
qn =
sn =
L 0 1 nπ x 1 nπ x β π ( x )cos dx = ( 2 m − 1 ) cos ∫ dx = 0 ∫ L − L L L L −L
0 1L ( 2m − 1 ) nπ x 1 nπ x π dx ( 2 m 1 ) sin = − {1 − ( −1)n} ∫ β ( x )sin dx = − ∫ L − L L n L L −L
Conclusion: Hence the Fourier Series of log(| sin x |) in (-L, L), 0 < L <π , x 0, |x|<π is ∞ ( − 1 )k 2 2 k B2 k L2 k ( 2 m − 1 )π log(sin L ) − ∑ ( +i ( 2 k + 1 )! 2 k =0
−
2
∞
1
∞
( −1 )λ −1( nπ )2 λ −1
( −1 )λ ( 2 L )2λ B2 λ λ ( − 1 )k nπ x ( cos ∑ 2 k −1 2λ ( 2λ − 2k + 1 )! L =1 k =1 ( nπ )
∞
( + ( −1 ) ∑ ∑ n ∑ π ( 2λ − 1 )( 2λ − 1 )! λ λ n =1
=1
n
1 − ( −1 )n nπ x sin n L n=0 ∞
−i( 2m − 1 )∑
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
Rene Ranjan William et al., J. Comp. & Math. Sci. Vol. 1 (6), 766-768 (2010)
Properties of the Fourier Coefficients: 1) a0 is a complex number 2) an is purely real 3) bn is purely imaginary REFERENCES 1. Advanced Engineering Mathematics by Erwin Kreyszig, 8th edition, John Wiley & sons Ltd, (1999). 2. CRC standard Mathematical tables and formulae by Daniel Zwillinger, 30th edition, C R C press, (1996).
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3. Advanced Engineering Mathematics by Louis C. Barrett, 6th edition, McGraw Hill (1995). 4. E.C. Titchmarsh, The Theory ofFunctions, Oxford University Press, London, (1961). 5. Y. Okuyama, Absolute Summability of Fourier Series and Orthogonal Series, Lecture Notes in Mathematics, Springer, Berlin, Vol. 1067 (1984). 6. H. Bor, A study on local properties of Fourier series, Nonlinear Anal. 57, 191â&#x20AC;&#x201C; 197 (2004). 7. G.H. Hardy, Divergent Series, Oxford University Press, Oxford, (1949).
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)