In plane regions, functions {entire function} can be single-valued and have no singularities.
Functions {greatest integer function} can have range equal nearest lower integer when domain is a real number.
Functions {identity function} can have range equal to domain.
As independent variable approaches value, function {infinite function} can approach +infinity, -infinity, or (infinity)^-1 = 0.
Functions that have one range value for each domain value and one domain value for each range value {bijective} can exchange domain and range {inverse function}. Reciprocals of relations can be relations {inverse relation}: x * R^1 * y = y * R^-1 * x. In the plane, inverse-function graphs reflect function graphs around line at 45-degree angle to x-axis and y-axis.
At points, inverses of single-valued functions can have multiple values {multiple-valued function} {multivalued function}. Multiple-valued functions are not functions.
Functions {postage stamp function} can have intervals with higher or lower constant range. Graphs look like stair steps.
Functions {prime function} can be a sum from k = 1 to k = s of k^(-s), where p is prime number, and s is positive number. It equals product from p = 1 to p = s of (1 - p^(-s))^(-1).
Number or expression can have inverse {reciprocal}: 1/x.
Functions {Bessel function} can have values P0(x) = 1, P1(x) = x, ..., PN(x) = ?, where P is function over interval, N is positive integer representing interval, and x is domain value. Bessel functions include beta functions.
Gamma of z plus w can divide into gamma of z times gamma of w {beta function}: beta(w,z) = (gamma(z) * gamma(w)) / gamma(z + w). Beta functions are Bessel functions. Beta of z and w equals beta of w and z. Beta function equals integral from x = 0 to x = 1 of x^(z - 1) * (1 - x)^(w - 1) * dx.
Holomorphic-function value at origin {Cauchy formula} is (1 / (2 * pi * i)) * contour integral (f(z)/z) * dz. Holomorphic-function value at point p is (1 / (2 * pi * i)) * contour integral (f(z)/(z - p)) * dz. Holomorphic-function nth-derivative value at origin is (n! / (2 * pi * i)) * contour integral (f(z)/z^(n + 1)) * dz.
Functions {complex function} {complex-valued function} can use complex variables: F(z), where z is complex variable. Complex function has form M + i*N, where M and N are real expressions, and i is imaginary number. For complex function F(z) = M + i*N, F(z') = M - i*N, where z' is z complex conjugate {fundamental theorem of complex numbers}.
Complex-number analytic functions {elliptic function} can be doubly periodic, be single-valued, have poles, and have singularity at infinity. Elliptic functions are power-series-equation solutions. They are elliptic-integral inverses. They are complex functions, because period ratios cannot be real numbers.
Weierstrass
Differential equations can have form dx^2 / dt^2 = A + B * x + C * x^2, where x is complex number. It has integral solutions {Weierstrass elliptic function}.
Jacobi
Differential equations can have form dx^2 / dt^2 = A + B * x + C * x^2 + D * x^3, where x is complex number. It has integral solutions {Jacobi elliptic function}.
Elliptic functions {Jacobian functions} can have higher powers. Determinants give solutions.
Darboux
Integrals {elliptic integral} {Darboux integral} can be (integral of P(x)) / (R(x))^0.5, where P is rational function, R is fourth-degree polynomial, and x is complex variable.
Abelian
Elliptic functions {Abelian integral} can have integral additions. They can also have algebraic and logarithmic terms. Integral of (R(u, z)) * dz, where f(u, z) = 0, requires more than one integral to describe domain. For Abelian integrals, equation genus is number of integrals needed to express solution {Abel's theorem}.
Complex analytic functions {holomorphic function} {holomorphic map} {differential function} {complex differentiable function} {regular function} can be differentiable for all derivatives.
equations
Complex-function real-part derivative with variable real part can equal complex-function imaginary-part derivative with variable imaginary part. Complex-function real-part derivative with variable imaginary part can equal negative of complex-function imaginary-part derivative with variable real part {Cauchy-Riemann equations, holomorphic function}. If Cauchy-Riemann equations hold, integrating on any path between two real complex-plane points obtains same result.
holomorphy
Paths can be deformable into each other or not {holomorphy}. Deformability allows canceling by going over same portion in opposite directions {homology, holomorphy}. Non-deformability does not allow canceling {homotopy, holomorphy}. If path has singularity, different paths are not homologous. At infinitesimal limit, holomorphic functions can be conformal, non-reflective, and orientation-preserving, so the only transformations are additions and multiplications. Reciprocal functions are holomorphic. Transformations (a*z + b)/(c*z + d) {bilinear transformation} {Möbius transformation, holomorphy} are holomorphic. Laplace equation in two dimensions has holomorphic function solutions.
Difference between holomorphic-function positive-frequency part and negative of negative-frequency part makes a function {hyperfunction}. Hyperfunctions have sums, derivatives, and products with analytic functions. Two hyperfunctions have no product. Hyperfunctions can represent Heaviside step function, Dirac delta function, and all analytic and holomorphic functions.
Multiple-valued complex functions {Riemann surface} can be complex-plane spirals {winding space, spiral}.
point
Riemann surfaces have a central point {branch point} about which to turn.
infinity
Surfaces can rejoin after a finite number of turns {finite order, surface} or can be infinite.
logarithm function
Logarithm-function Riemann surface is not compact but can compact to Riemann sphere.
Riemann sphere
The simplest compact/closed Riemann surface {Riemann sphere} has complex plane goes through equator. Complex plane stereographically projects onto one hemisphere, and its reciprocal projects onto other hemisphere. Circles or straight lines on complex planes are circles on spheres.
genus
Sphere has genus 0, because it has no complex moduli and has three holomorphic self-transformation parameters, for bilinear transformations. Torus has genus 1, because it has one complex modulus and one holomorphic self-transformation parameter, for translation. Genus 2 has three complex moduli and no holomorphic self-transformation parameters. Genus n has 3*n - 3 complex moduli and no holomorphic self-transformation parameters.
If Riemann hypothesis is true, functions {Riemann zeta function} can find number of primes less than N. Riemann zeta function is Dirichlet series. For complex numbers, zeta(z) = 1^-z + 2^-z + 3^-z + ..., which converges if z real part > 1, zeta(z) = 0, and z = -2, -4, -6, ... If imaginary numbers are input to zeta function, output can equal 0. Riemann zeta function equals infinity if z = 0 or 1.
Riemann hypothesis
Riemann zeta function converges if z real part = -0.5 {Riemann hypothesis} {Riemann problem}. This has no proof yet.
primes
If Riemann hypothesis is true, equation-zero locations give prime-number locations.
properties
For numbers x and N, zeta(x) = 1 + 1/2^x + 1/3^x + ... + 1/N^x. If x = 1, zeta is harmonic series. For x = 2, zeta converges to (pi)^2/6 [Euler, 1748], so sum of rational fractions gives transcendental number.
Elliptic functions can have inverses {theta function, elliptic}. Sum from z = -infinity to z = +infinity of e^(-t * n^2 + 2 * n * i * z), where n is 0 or positive integer, and t is parameter. If t > 0, theta has real part.
Multiple-valued complex functions can be sheets or spirals {winding space, function} in complex-plane Riemann surfaces.
Over an interval, at domain points, function {continuous function} limits can equal range values. All polynomials are continuous functions. At points, if two functions are continuous, their sums, products, and quotients are continuous.
Over an interval, at domain points, function {discrete function} limits can not equal range values. Functions can have values only at a finite number of points. Finite strings can represent finite discrete point sets, so discrete functions can map finite strings onto finite strings.
Flexible heavy cable hanging from two ends makes a curve {catenary} with function y = a * cosh(x/a). y = a if x = 0.
Functions {lacunary} can have natural boundaries.
Functions {error function} can have value (2 / (pi^0.5)) * integral from t = 0 to t = z of e^(-t^2) * dt.
Functions {exponential function, power} can have constant base {radix, base} {base, radix} raised to variable power: 10^x and e^x. exp(x) equals e^x if base is e or equals 10^x if base is 10. Exponential functions are logarithmic-function inverses.
Functions {Laplace transform} can be integral from t = 0 to t = +infinity of e^(-s*t) * F(t) * dt. If function value is 1/s, then F(t) = 1. Integral from x = 0 to x = infinity of integral of y = 0 to y = infinity of e^(-u*x - v*y) * F(x,y) * dx * dy is Laplace transform.
Functions {logarithmic function, exponential} can use variable powers of constant bases. ln(x) depends on base e {natural logarithm, base}. log(x) depends on base 10 {common logarithm}. Logarithms are exponential inverses.
Taking logarithm of value gives exponent to use with base. log(100) = x = log(10^x) = 2. ln(e^x) = x.
product
Product logarithm equals sum of factor logarithms {law of exponents}: log(M * N) = log(M) + log(N).
Exponential-function product equals exponential function of exponent sum: exp(x) * exp(y) = exp(x + y).
hyperbola
Logarithm equals area under hyperbola integrated from 1 to y, because hyperbola has equation y = 1/x, and integral of 1/y is logarithm.
line
Exponential equals 1 + n + n^2 / 2, for value n. Therefore, exponential equals 1 plus value plus area under line plus area under line at 45-degree angle.
Filtering functions {comb function} {shah function} {sampling function} can have unit amplitude, zero width, and unit area at regular intervals: _|_|_|_|_. Comb filter selects points or intervals.
Filtering functions {Dirac delta function} {pulse filter} can have infinite amplitude, zero width, and unit area: __|__. The pulse filter selects point or interval.
Functions {filter function} can sample another function at intervals, to measure spatial wave-front spectrum or reveal pattern shape and size.
Filtering functions {rectangle function} {pi function} can equal one over interval and equal zero elsewhere: __-__. Filter selects point or interval.
Filtering functions {triangle function} can equal zero except over interval where it rises then falls linearly at 45-degree angle: __A__. Filter selects interval and weights middle most and ends least.
Functions {gamma function} can be gamma(n) = (n! * n^z) / (z * (z + 1) * (z + 2) * ... * (z + n)), where n is number. It has limit at infinity. Gamma function is integral {Euler's integral} from t = 0 to t = infinity of t^(z - 1) * e^(-t) * dt. Value at z + 1 is z! = product from i = 1 to i = z of integral from x = 0 to x = 1 of (-log(x))^z * dx.
Functions {psi function} {digamma function} can be derivatives of gamma divided by gamma, so value at z + 1 is value at z plus 1 divided by z: psi(z + 1) = psi(z) + 1/z.
Functions {even function} can not change sign or absolute value when independent-variable sign changes. Even-function graphs are symmetric to y-axis.
Functions {odd function} can change sign when independent-variable sign changes. Odd-function graphs are symmetric to origin.
Forms {form, polynomial} are polynomial expressions. Forms can have any variable degree or number {module, polynomial} {modular system}. Forms have finite numbers of basic forms {basis, polynomial}, which make complete systems {Hilbert's basis theorem, form}.
In polynomials, variable has highest exponent {degree, polynomial} {index, polynomial}. For example, polynomial x^2 + 4*x has degree 2.
For one-variable functions, variable power determines graph {graphing}.
Linear function has first power and is straight line, with slope and y-intercept.
Quadratic function has second power and is parabola, with two x-intercepts and one y-intercept.
Cubic equation has third power and has S shape, with three x-intercepts and one y-intercept.
Products of independent and dependent variables, x*y, are hyperbolas.
conics
General functions can add two variables, each raised to second and first power: A * x^2 + B * y^2 + C * x + D * y + E. Functions can equal zero. Equations are ellipses if b^2 - 4*a*c < 0, hyperbolas if b^2 - 4*a*c > 0, and parabolas if b^2 - 4*a*c = 0.
If two polynomials are equal, then coefficients of terms with same variables with same exponents are equal {undetermined coefficients principle} {principle of undetermined coefficients}.
To add polynomials {polynomial addition}, first put terms in simplest form. Then add coefficients of terms that have same variables with same exponents. Sums have same number of terms as total number of different terms in both polynomials.
polynomial division
To divide two polynomials, write polynomials with terms decreasing from highest exponent term. Divide smaller second-polynomial first term into larger first-polynomial first term. Multiply smaller polynomial by quotient. Subtract product from first polynomial. Divide difference by second-polynomial first term, to get new quotient. Then repeat steps. For example, (12*x^2 - x - 6) / (3*x + 2) = (4*x) * (3*x + 2) - 9*x - 6 = (4*x - 3) * (3*x + 2).
polynomial multiplication
To multiply polynomials, multiply each first-polynomial term by each second-polynomial term. Product-term number is number of first-polynomial terms times number of second-polynomial terms. Put terms in simplest form. Add coefficients of terms that have same variables with same exponents.
Products can result in ordered pairs {Cartesian product, function}, denoted [X,Y].
A number or polynomial {factor, polynomial} can divide into another number or polynomial with no remainder. Try to find prime number that factors, try coefficient, try variable, and then try simple polynomial. Different terms can share a prime-factor product {greatest common factor, polynomial}, which divides into terms with no fractional remainder. Linear or quadratic polynomial with real coefficients can have factors with real coefficients.
Polynomials can equal smaller-polynomial products {factoring, polynomial}. For example, a^3 - b^3 factors to (a - b)*(a^2 + a*b + b^2).
difference of squares
Binomials can factor if it they are differences between two squares. For example, x^2 - y^2 factors as (x + y)*(x - y). 9*(x^2) - 64*(y^4) factors as (3*x + 8*(y^2))*(3x - 8*(y^2)).
process
To factor polynomial, first try to find number, coefficient, or variable {monomial factor} that is in all terms. Then try factor with two terms {binomial factor}. First, try binomial whose first term has coefficient that factors highest-power-term coefficient and has highest-power-term variable with no exponent. Second term is number that factors polynomial number term.
process: quadratic trinomial
To factor quadratic trinomials, first place terms in decreasing order of powers. Factor trinomial by highest-power-term coefficient. Try to factor trinomial by variable. Find constant-term numerator and denominator factors. From factors, use two numbers that add to middle-term coefficient. Then factors are (x + number1) and (x + number2).
a*(x^2) + b*x + c factors to a*(x^2 + x*(b/a) + c/a) which factors to (x + c1/a1)*(x + c2/a2), where c = c1*c2, a = a1*a2, b/a = (c1/a1 + c2/a2), and b = c1*a2 + c2*a1.
process: quadrinomial
To factor quadrinomials, first try to find a monomial factor using any term pair. For example, a + b + c + d can factor to e*(f + g) + c + d. Then try to find binomial factor shared by two term pairs. For example, 6*a*x - 2*b - 3*a + 4*b*x factors to 3*a*(2*x - 1) + 2*b*(2*x - 1) which factors to (2*x - 1)*(3*a + 2*b).
process: test binomial
If polynomial has no factors, use test binomial factor. The variable is in highest polynomial term with no exponent. Add constant. For example, x^2 + x + 1 has test factor (x + 1). Divide polynomial by test factor {synthetic division}, to get quotient polynomial and remainder polynomial {remainder theorem}. For example, (x^2 + x + 1)/(x + 1) = x + 1/(x^2 + x + 1).
If remainder is zero, test factor is polynomial factor {factor theorem}. If remainder is zero, negative of constant is a polynomial zero {converse, factor theorem}. For example, (x^2 + 2x + 1)/(x + 1) = x + 1, so remainder is zero, and x is -1.
After dividing power functions or polynomials by modulus, if remainders are the same, the power functions or polynomials are congruent quadratics {quadratic reciprocity law} {law of quadratic reciprocity}, biquadratics {law of biquadratic reciprocity}, and cubics {cubic reciprocity law} {law of cubic reciprocity}.
Functions {homogeneous function} can allow factoring a constant: (k^n) * f(x, y, ...) = f(k*x, k*y, ...), where n is degree, k is constant, f is homogeneous function, and x, y, ... are variables.
Functions {linear function, polynomial} can relate variables with first power.
Variable base can have constant power in function {power function}|. For example, x^3.
Finite complete systems for any-degree binary forms can have rational integral invariants and covariants {Clebsch-Gordan theorem}. Covariants are function projections in binary form.
Forms can have any variable degree or number. Forms can use finite numbers of basic forms, which make complete systems {Hilbert's basis theorem, polynomial} {Hilbert basis theorem, polynomial}.
For all terms, sums of variable exponents can be constant {homogeneous expression}. An example is a^2 * b^1 + a^1 * b^2.
If denominators have no variables, all variable exponents are positive {integral expression}.
Polynomials {perfect power, polynomial} can be nth powers of similar polynomials.
Binomials squared make trinomials {perfect trinomial square}.
Algebraic polynomials {quantic} can have two or more variables, be homogeneous, and be rational integral functions. Quantics can have two variables {binary polynomial}, three variables {ternary polynomial}, four variables {quaternary polynomial}, two degrees {quadric polynomial}, three degrees {cubic polynomial}, four degrees {quartic polynomial}, or five orders or degrees {quintic polynomial}.
Expressions {radical expression} can contain radical signs.
Expressions can have no radical expressions or fractional exponents {rational expression}. Rational expressions can be quotients of two polynomials. Polynomials are rational integral expressions.
Trigonometry is about ratios of right-triangle sides and acute angles {trigonometric function, mathematics}.
triangle sides
Right triangle has longest side opposite right angle {hypotenuse, right triangle}, side opposite acute angle {opposite side, right triangle}, and side adjacent to acute angle {adjacent side}.
ratios
Acute right-triangle angles have ratios of opposite side to hypotenuse {sine}, adjacent side to hypotenuse {cosine}, opposite side to adjacent side {tangent, angle}, adjacent side to opposite side {cotangent}, hypotenuse to opposite side {cosecant}, and hypotenuse to adjacent side {secant, trigonometry}.
sin = opposite/hypotenuse. cos = adjacent/hypotenuse. tan = opposite/adjacent. csc = hypotenuse/opposite. sec = hypotenuse/adjacent. cot = adjacent/opposite.
trigonometric relations
Tangent equals sine divided by cosine: tan = sin/cos. Cotangent equals cosine divided by sine: cot = cos/sin.
Sine equals cosecant reciprocal: sin = 1/csc. Cosine equals secant reciprocal: cos = 1/sec. Tangent equals cotangent reciprocal: tan = 1/cot. Cotangent equals tangent reciprocal: cot = 1/tan. Secant equals cosine reciprocal: sec = 1/cos. Cosecant equals sine reciprocal: csc = 1/sin.
domain and range
For all trigonometric functions, domain is all real numbers.
The sine and cosine range is from negative one to positive one. The secant range is from positive one to infinity. Cosecant range is from negative one to negative infinity. Tangent and cotangent range is all real numbers.
angles
Trigonometric functions can have angles of less than 0 or more than 90 degrees. Trigonometric functions can have angles between 270 and 360 degrees and negative acute angles. sin(A) = -sin(360 - A). tan(A) = -tan(360 - A). csc(A) = -csc(360 - A). cos(A) = cos(360 - A). cot(A) = -cot(360 - A). sec(A) = sec(360 - A). Trigonometric functions can have angles between 180 and 270 degrees and negative obtuse angles. sin(A) = -sin(A - 180). tan(A) = tan(A - 180). csc(A) = -csc(A - 180). cos(A) = -cos(A - 180). cot(A) = cot(A - 180). sec(A) = -sec(A - 180). Trigonometric functions can have obtuse angles between 90 and 180 degrees. sin(A) = sin(180 - A). tan(A) = tan(180 - A). csc(A) = csc(180 - A). cos(A) = -cos(180 - A). cot(A) = -cot(180 - A). sec(A) = -sec(180 - A).
angles: radians
Angle 360 degrees = 2*pi radians.
angles: phase
To make angle be from 0 to 360 degrees, add or subtract multiple of 360 degrees = 2 * pi radians. All trigonometric functions repeat values for angle plus 2 * n * pi and angle minus 2 * n * pi, where n is integer. For example, sin(A) = sin(A + 2 * n * pi) and sin(A) = sin(A - 2 * n * pi).
angles: differences
Trigonometric angle-difference functions relate to trigonometric angle functions. sin(A - B) = sin(A) * cos(B) - cos(A) * sin(B). cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B).
angles: negative
Trigonometric negative-angle functions relate to trigonometric positive-angle functions. sin(-A) = -sin(A). cos(-A) = cos(A). tan(-A) = -tan(A).
angles: sums
Trigonometric angle-sum functions relate to trigonometric angle functions. sin(A + B) = sin(A) * cos(B) + cos(A) * sin(B), so sin(2*A) = 2 * sin(A) * cos(A). cos(A + B) = cos(A) * cos(B) - sin(A) * sin(B), so cos(2*A) = (cos(A))^2 - (sin(A))^2 = 2 * (cos(A))^2 - 1. tan(2*A) = (2 * tan(A)) / (1 - (tan(A))^2). tan(A) = (1 - cos(2*A)) / sin(2*A) = sin(2*A) / (1 + cos(2*A)).
sums and products
Sums of trigonometric functions relate to products of trigonometric functions. sin(A + B) + sin(A - B) = 2 * sin(A) * cos(B). sin(A + B) - sin(A - B) = 2 * cos(A) * sin(B). cos(A + B) + cos(A - B) = 2 * cos(A) * cos(B). cos(A - B) - cos(A + B) = 2 * sin(A) * sin(B). Set A = (x + y) / 2 and B = (x - y) / 2 to solve for sin(x) + sin(y), sin(x) - sin(y), cos(x) + cos(y), or cos(y) - cos(x).
Trigonometric functions relate to unit circle {circular function}.
Cosecant minus one is trigonometric function {excosecant} (excsc).
Secant minus one is trigonometric function {exsecant} (exsec).
Functions {hyperbolic function, trigonometry} can relate to unit hyperbola as trigonometric functions relate to unit circle. For independent variable x and base e of natural logarithms, (e^x - e^-x) / 2 {hyperbolic sine} (sinh). (e^x + e^-x) / 2 {hyperbolic cosine} (cosh). (e^x - e^-x) / (e^x + e^-x) {hyperbolic tangent} (tanh). (e^x + e^-x) / (e^x - e^-x) {hyperbolic cotangent} (coth). 2 / (e^x + e^-x) {hyperbolic secant} (sech). 2 / (e^x - e^-x) {hyperbolic cosecant} (csch).
relations
(sinh(x))^2 + (cosh(x))^2 = cosh(2*x). (cosh(x))^2 - (sinh(x))^2 = 1. sinh(x + y) = sinh(x) * cosh(y) + cosh(x) * sinh(y). cosh(x + y) = cosh(x) * cosh(y) + sinh(x) * sinh(y). arcsinh(x) = ln(x + (x^2 + 1)^0.5).
Trigonometric functions {principal branch function} can have domain 0 to 90 degrees, for acute angles.
If angle is acute, between 0 and 90 degrees, trigonometric functions have inverses {trigonometric inverse}: sin^-1 {arcsine}, cos^-1 {arccosine}, tan^-1 {arctangent}, cot^-1 {arccotangent}, sec^-1 {arcsecant}, and csc^-1 {arccosecant}.
One minus cosine is function {versed sine} {versine}. One minus the sine is function {coversed sine} {versed cosine} {coversine}.
Functions {wave function} can be waves.
series
Periodic functions can be trigonometric series. If period is T, series is: a0 + sum over i of (ai * cos(2n * pi * tau / T) + bi * sin(2n * pi * tau / T)), where a0 = (1/T) * (integral from -T/2 to T/2 of f(t) * dt), ai = (2/T) * (integral of f(t) * cos(2n * pi * t / T)), and bi = (2/T) * (integral of f(t) * sin(2n * pi * t / T)).
Sine or cosine can be zero. Even periodic function uses cosine. Odd periodic function uses sine.
period
For function over interval with width x, period T is twice interval length x: T = 2*x.
jump
Term coefficients depend on differences {jump} between left-hand and right-hand function limits, derivatives at jump points, and second derivatives at jump points.
analyzer
Harmonic analyzer can find first 20 coefficients from function graph and areas. Integrator circuits can calculate area.
Fourier trigonometric series {Fourier transform, function} can represent function over interval.
transformation
Function can transform by multiplying function by periodic function and integrating. Integral from x = -infinity to x = +infinity of g(x) * e^(-i * 2 * pi * u * x) * dx, where x = domain value, g(x) is function, u is frequency, and i is square root of -1.
transformation: coordinates
Fourier trigonometric series can transform coordinates. (1/(2 * pi)) * (integral from q = -infinity to q = +infinity of (e^(i*q*x)) * dq) * (integral from a = -infinity to a = +infinity of F(a) * (e^(-i*q*x)) * da).
limit
Fourier series go to limit as period approaches infinity.
time
Time t relates to phase A: t = tan(A/2).
power series
Complex power series can represent periodic functions with holomorphic positive and negative frequency.
Mechanisms {harmonic analyzer} can find first 20 Fourier-transform coefficients, using function graphs and measuring areas.
Functions {Fresnel integral} can have value C(z) = integral, from t = 0 to t = z, of cos(pi * t^2 / 2) * dt. It equals 0.5 if z equals infinity.
Functions {Gudermannian function} Gdx can have sin(u) = tanh(x), cos(u) = sech(x), or tan(u) = sinh(x) (Christoph Gudermann) [1798 to 1852].
One-half versed sine is a function {haversine} (hav).
Functions {Legendre function} S(z) can be integrals from t = 0 to t = z of sin((pi * t^2)/2) * dt and equal 0.5 if z equals infinity.
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Date Modified: 2022.0225