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Mathematical Functions…But what about the functions themselves? … Plots of some standard mathematical functions. The top row shows three trigonometric functions.
Mathematical Functions…The basic definition of this function is fairly simple. … A curve associated with the so-called Riemann zeta function. The zeta function Zeta[s] is defined as Sum[1/k s , {k, ∞ }] .
Mathematical Functions…When α is a square root, then as discussed in the previous section , the continued fraction representation is purely repetitive, Curves obtained by adding together various sine functions.
Mathematical Functions…[No text on this page] Curves obtained by adding or subtracting exactly two sine or cosine functions turn out to have a pattern of axis crossings that can be reproduced by a generalized substitution system.
Note (a) for Mathematical Functions…Mathematical functions (See page 1091 .) … Other standard mathematical functions that oscillate at large x include JacobiSN and MathieuC . Most hypergeometric-type functions either increase or decrease exponentially for large arguments, though in the directions of Stokes lines in the complex plane they can oscillate sinusoidally.
Note (c) for Mathematical Functions…Two sine functions Sin[a x] + Sin[b x] can be rewritten as 2 Sin[1/2(a + b) x] Cos[1/2(a - b) x] (using TrigFactor ), implying that the function has two families of equally spaced zeros: 2 π n/(a + b) and 2 π (n + 1/2)/(b - a) .
Note (b) for Mathematical Functions…Lissajous figures Plotting multiple sine functions each on different coordinate axes yields so-called Lissajous or Bowditch figures, as illustrated below. If the coefficients inside all the sine functions are rational, then going from t = 0 to t = 2 π Apply[LCM, Map[Denominator, list]] yields a closed curve.
Note (f) for Mathematical Functions…Three sine functions All zeros of the function Sin[a x] + Sin[b x] lie on the real axis.
Note (h) for Mathematical Functions…Many sine functions Adding many sine functions yields a so-called Fourier series (see page 1074 ). … Other so-called Fourier series in which the coefficient of Sin[m x] is a smooth function of m for all integer m yield similarly simple results.
Note (d) for Mathematical Functions…Differential equations [for sine sums] The function Sin[x] + Sin[ √ 2 x] can be obtained as the solution of the differential equation y''[x] + 2 y[x] - Sin[x]  0 with the initial conditions y[0]  0 , y'[0]  2 .
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