Three sine functions All zeros of the function Sin[a x] + Sin[b x] lie on the real axis. But for Sin[a x] + Sin[b x] + Sin[c x] , there are usually zeros off the real axis (even say for a = 1 , b = 3/2 , c = 5/3 ), as shown in the pictures below. … But in a case like Sin[x] + Sin[ √ 2 x] + Sin[ √ 3 x] there is a continuous distribution of spacings between zeros, as shown on a logarithmic scale below.

Intrinsically defined curves With curvature given by a function f[s] of the arc length s , explicit coordinates {x[s], y[s]} of points are obtained from (compare page 1048 ) NDSolve[{x'[s]  Cos[ θ [s]], y'[s]  Sin[ θ [s]], θ '[s]  f[s], x[0]  y[0]  θ [0]  0}, {x, y, θ }, {s, 0, s max }] For various choices of f[s] , formulas for {x[s], y[s]} can be found using DSolve : f[s] = 1: {Sin[ θ ], Cos[ θ ]} f[s] = s: {FresnelS[ θ ], FresnelC[ θ ]} f[s] = 1/ √ s : √ θ {Sin[ √ θ ], Cos[ √ θ ]} f[s] = 1/s: θ {Cos[Log[ θ ]], Sin[Log[ θ ]]} f[s] = 1/s 2 : θ {Sin[1/ θ ], Cos[1/ θ ]} f[s] = s n : result involves Gamma[1/n, ±  θ n/n ] f[s] = Sin[s] : result involves Integrate[Sin[Sin[ θ ]], θ ] , expressible in terms of generalized Kampé de Fériet hypergeometric functions of two variables. … The case of f[s] = a Sin[b s] was studied by Eduard Lehr in 1932. Cases related to f[s] = s Sin[s] were studied by Alfred Gray around 1992 using Mathematica.

And as the pictures on the facing page indicate, for any curve like Sin[x] + Sin[ α x] the relative arrangements of these crossing points turn out to be related to the output of a generalized substitution system in which the rule at each step is obtained from a term in the continued fraction representation of ( α – 1)/( α + 1) .

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 .

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) .

The pictures below show Sum[Sin[n x]/n, {n, k}] for various numbers of terms k . … The pictures below show Sum[Sin[n 2 x]/n 2 , {n, k}] , where in effect all coefficients of Sin[m x] other than those where m is a perfect square are set to zero. … Note that for x of the form p π /q , the k = ∞ sum is just ( π /q/(2q)) 2 Sum[Sin[n 2 p π /q]/Sin[n π /(2q)] 2 , {n, q - 1}] The pictures below show Sum[Cos[2 n x], {n, k}] (as studied by Karl Weierstrass in 1872).

[Mollusc] shell model The center of the opening of a shell is taken to trace out a helix whose {x, y, z} coordinates are given as a function of the total angle of revolution t by a t {Cos[t], Sin[t], b} . … The complete surface of the shell is obtained by varying both t and θ in a t {Cos[t] (1 + c (Cos[e] Cos[ θ ] + d Sin[e] Sin[ θ ])), Sin[t] (1 + c (Cos[e] Cos[ θ ] + d Sin[e] Sin[ θ ])), b + c (Cos[ θ ] Sin[e] - d Cos[e] Sin[ θ ])} where c varies from 0.4 to 1.6 on row (c), d from 1 to 4 on row (d) and e from 0 to 1.2 on row (e).

If one allows trigonometric functions, any equation for integers can be converted to one for real numbers; for example x 2 + y 2  z 2 for integers is equivalent to Sin[ π x] 2 + Sin[ π y] 2 + Sin[ π z] 2 + (x 2 + y 2 - z 2 ) 2  0 for real numbers.

Recognizing repetition [in sounds] The curve of the function Sin[x] + Sin[ √ 2 x] shown on page 146 looks complicated to the eye.

BesselJ[0, x] goes like Sin[x]/ √ x for large x while AiryAi[-x] goes like Sin[x 3/2 ]/x 1/4 .