ENTRY FUNCTIONS Authors: Oliver Knill: 2003, Literature: no +------------------------------------------------------------ | real-valued function +------------------------------------------------------------ A real-valued function is usually assumed to be map to the reals. +------------------------------------------------------------ | abscissa +------------------------------------------------------------ abscissa The x-coordinate in an (x,y) graph of a function. The y-coordinates is called ordinate. +------------------------------------------------------------ | ordinate +------------------------------------------------------------ ordinate The y-coordinate in an (x,y) graph of a function. The x-coordinates is called abscissa. +------------------------------------------------------------ | Airy function +------------------------------------------------------------ The Airy function is defined as the solution of the differential equation y''-x y =0. +------------------------------------------------------------ | Briggsian logarithm +------------------------------------------------------------ The Briggsian logarithm also called common logarithm is the logarithm to the base 10. +------------------------------------------------------------ | Bessel function +------------------------------------------------------------ THe Bessel function is a special function. Bessel function of the first kind of order zero is defined as J_0 = sum_k=0^infinity (-1)^k (x/2)^2k/(k!)^2. +------------------------------------------------------------ | Sin +------------------------------------------------------------ The Sin is a trigonometric function. It can be defined by its series sin(x) = x-x^3/3! + x^5/5! - ..., where 5!=5 4 3 2 1 is the factorial of 5. The sine function can also be defined as the imaginary part of exp(i x) = cos(x) + i sin(x), where i =(-1)^(1/2) is the imaginary unit. Examples of values sin(0)=0, sin(pi/2)=1, sin(pi)=0, sin(3 pi/2)=-1. +------------------------------------------------------------ | Csc +------------------------------------------------------------ Csc The cosecant is defined as csc(x)=1/sin(x). +------------------------------------------------------------ | Arcsin +------------------------------------------------------------ Arcsin The arcsin is the inverse of sin. It is also denoted by sin^(-1)(x) or asin(x). One has the identities arcsin(sin(x))=x, or sin(arcsin(x))=x. +------------------------------------------------------------ | Sinh +------------------------------------------------------------ Sinh The hyperbolic sine can be defined as sinh(x)=(exp(x)-exp(-x))/2. Examples: sinh(0)=0. +------------------------------------------------------------ | ArcSinh +------------------------------------------------------------ ArcSinh The inverse of sinh is called arcsinh. +------------------------------------------------------------ | Cos +------------------------------------------------------------ The trigonometric function Cos can be defined by its series cos(x) = 1-x^2/2!+x^4/4! - x^6/6! - ..., where 4!=4 3 2 1 is the factorial of 4. It can also be defined as the real part of exp(i x), where i =(-1)^1/2 is the imaginary unit, the square root of -1. Examples: cos(0)=1, cos(pi/2)=0 sin(pi)=-1, sin(3pi/2)=0. +------------------------------------------------------------ | Arccos +------------------------------------------------------------ Arccos The inverse of the function cos is written arccos(x), also denoted by cos^(-1)(x) or acos(x). We have the identities arccos(cos(x))=x, or cos(arccos(x))=x. +------------------------------------------------------------ | Sec +------------------------------------------------------------ Sec The secant is defined as sec(x)=1/cos(x). +------------------------------------------------------------ | Cosh +------------------------------------------------------------ Cosh The hyperbolic cosine can be defined as sinh(x)=(exp(x)+exp(-x))/2. Examples: cosh(0)=1. +------------------------------------------------------------ | ArcCosh +------------------------------------------------------------ ArcCosh The inverse of cosh is called arccosh. +------------------------------------------------------------ | Tan +------------------------------------------------------------ The Tan is a trigonometric function. It can be defined as tan(x)=sin(x)/cos(x). Examples: tan(0)=0, tan(pi/4)=1. +------------------------------------------------------------ | Arctan +------------------------------------------------------------ Arctan The inverse of tan is the function arctan(x). It is also called tan^(-1)(x). One has arctan(tan(x))=x and tan(arctan(x))=x. Examples: arctan(1) = pi/2. +------------------------------------------------------------ | Cot +------------------------------------------------------------ Cot is a trigonometric function. It can be defined as cot(x)=cos(x)/sin(x). It can also be defined geometrically as the relation of two sides in a right angle triangle if x is one of the angles. Examples: cot(pi/2)=0, cot(pi/4)=1. +------------------------------------------------------------ | Exp +------------------------------------------------------------ Exp is the exponential function. It can be defined by its series exp(x) = 1+x+x^2/2!+x^3/3!+x^4/4!+... where 4!=4 3 2 1 is the factorial of 4. Examples: exp(0)=1, exp(1)=e=2.712.... +------------------------------------------------------------ | Sqr +------------------------------------------------------------ Sqr The square of a number is the product of the number by itself. For example, the square of 4 is 16. The square of a function sin(x) is denoted by sin^2(x). +------------------------------------------------------------ | Zeta +------------------------------------------------------------ Zeta zeta(s) is the Riemann zeta function. It is defined for complex numbers s which have Re(s)>1 as zeta(s)=1+1/2^s+1/3^s+.... The function can be continued to the entire complex plane except at s=1, where the function has a singularity. The zeta function has zeros at -2,-4,-6 and also zeros on the real line Re(s)=1/2. The famous Riemann hypothesis claims that all the nontrivial zeros are on this line. This conjecture remains unproven until today and is considered one of the most important open problems in mathematics. +------------------------------------------------------------ | Log +------------------------------------------------------------ Log The logarithm is the inverse to the exponential function: log(exp(x)) = x and exp(log(x)) = x. For example: log(1)=0, log(e)=1. The logarithm function satisfies for example the laws log(x y) = log(x) + log(y), log(x/y)=log(x)-log(y), log(x^y)=y log(x). +------------------------------------------------------------ | Sqrt +------------------------------------------------------------ Sqrt The square root of a number x is the number which This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 24 entries in this file. COUNT: 24