In mathematics, a trigonometric series is a series of the form:
\( {\displaystyle {\frac {A_{0}}{2}}+\displaystyle \sum _{n=1}^{\infty }(A_{n}\cos {nx}+B_{n}\sin {nx}).} \)
It is called a Fourier series if the terms \( A_{n} \) and \( B_{{n}} \) have the form:
\( A_{{n}}={\frac {1}{\pi }}\displaystyle \int _{0}^{{2\pi }}\!f(x)\cos {nx}\,dx\qquad (n=0,1,2,3\dots ) \)
\( B_{{n}}={\frac {1}{\pi }}\displaystyle \int _{0}^{{2\pi }}\!f(x)\sin {nx}\,dx\qquad (n=1,2,3,\dots ) \)
where f is an integrable function.
The zeros of a trigonometric series
The uniqueness and the zeros of trigonometric series was an active area of research in 19th century Europe. First, Georg Cantor proved that if a trigonometric series is convergent to a function f(x) on the interval \( [0,2\pi ]\), which is identically zero, or more generally, is nonzero on at most finitely many points, then the coefficients of the series are all zero.[1]
Later Cantor proved that even if the set S on which f is nonzero is infinite, but the derived set S' of S is finite, then the coefficients are all zero. In fact, he proved a more general result. Let S0 = S and let Sk+1 be the derived set of Sk. If there is a finite number n for which Sn is finite, then all the coefficients are zero. Later, Lebesgue proved that if there is a countably infinite ordinal α such that Sα is finite, then the coefficients of the series are all zero. Cantor's work on the uniqueness problem famously led him to invent transfinite ordinal numbers, which appeared as the subscripts α in Sα .[2]
References
[1]
Cooke, Roger (1993), "Uniqueness of trigonometric series and descriptive set theory, 1870–1985", Archive for History of Exact Sciences, 45 (4): 281–334, doi:10.1007/BF01886630.
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Sequences and series
Integer
sequences
Basic
Arithmetic progression Geometric progression Harmonic progression Square number Cubic number Factorial Powers of two Powers of three Powers of 10
Advanced (list)
Complete sequence Fibonacci numbers Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number
Fibonacci spiral with square sizes up to 34.
Properties of sequences
Cauchy sequence Monotone sequence Periodic sequence
Properties of series
Convergent series Divergent series Conditional convergence Absolute convergence Uniform convergence Alternating series Telescoping series
Explicit series
Convergent
1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ 1 + 1/2s+ 1/3s + ... (Riemann zeta function)
Divergent
1 + 1 + 1 + 1 + ⋯ 1 + 2 + 3 + 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) Infinite arithmetic series 1 − 2 + 3 − 4 + ⋯ 1 − 2 + 4 − 8 + ⋯ 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series
Taylor series Power series Formal power series Laurent series Puiseux series Dirichlet series Trigonometric series Fourier series Generating series
Hypergeometric
series
Generalized hypergeometric series Hypergeometric function of a matrix argument Lauricella hypergeometric series Modular hypergeometric series Riemann's differential equation Theta hypergeometric series
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