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# Function Theory 函数论

## Rules:

put function arguments in parentheses: sin(x) -- correct, sin x -- incorrect; do not omit multiplication: x*sin(x) -- correct, x sin(x) -- incorrect. x**y or pow(x,y) for power. Usual keywords are lowercase, which are different from uppercase, e.g. sin is different from Sin. Its default variable is small letter x, but its default index variable in discrete math is k.

## Function Search

• Search function with ? in the input box, e.g.

• search function in wiki. e.g.

• serach function in Digital Library of Mathematical Functions NIST, e.g.
erf

## List of Math function and operations

JavaScript math - function reference

## Classification 分类

### 三级学科

▪ 4110:实变函数论 ▪ 4120:单复变函数论 ▪ 4130:多复变函数论
▪ 4140:函数逼近论 ▪ 4150:调和分析     ▪ 4160:复流形
▪ 4170:特殊函数论 ▪ 4199:函数论其他学科

Category Functions
constantsE, infinity, inf, e, false, i, null, pi, true, oo
operator +, -, *, /, ^, **, >, >=, <, <=, ==, !=, <>
complex operator add(x,y), div(x,y), mul(x,y), sub(x,y), pow(x,y), root(x,y), inv(x), neg(x)
Algebra combine, factor, expand, inverse, lsolve, fsolve, simplify, solve, nsolve, rsolve, psolve, test, ftest, rtest
Arithmetic abs(x), cbrt(x), ceil(x), exp(x), fix(x), floor(x), gcd(x,y), hypot(x), lcm(x,y), log(x), log10(x), ln(x), norm(x), nthRoot(x), round(x), sign(x), sgn(x), sqrt(x),
Combinatorics bell, binomial, catalan, combination, Stirling1, Stirling2
complexarg(x), conj(x), im(x), imag(x), re(x), real(x)
Convert convert, toexp, tolog, totrig, totrigh, tocomplex, toint, tosum, toseries
Construction see mathjs
Calculus lim, limit, d, diff, int, integral, integrate, convolute, laplace, inverselaplace, fourier, semid, semiint, nthd, ode, pde, dsolve, pdsolve, odeplot, taylor, series
Discrete math difference differences Delta sum sums sum0n sum1n sum0inf sum1inf infsum partialsum psum infsum1 partialsum1 psum1 recurrence rsolve tosum
Expression eval, evaluate, help, ?
Geometrycircle(1), semicircle(1), oval(2,1), disc(1), ring(1,2), area, length, tangent, slope,
Logical and, not, or, xor, >, >=, <, <=, ==, !=, <>
matrix see mathjs
numeric n, nint, nsum, nsolve, see mathjs
Probability binomial(3,x), combination(3,x), factorial(x), factorial2(x), gamma(x), random(x), P(x>0.5)
polymonial bell(n,x), euler(n,x), binomial(n,k), bernoulli(n,x), Stirling1(n,x), Stirling2(n,x), harmonic(3,x), poly(3,x), hermite(3,x),
number theory bell(x), binomial(n,k), bernoulli(n,x), catalan(x), euler(n,x), fac(x), factorial(x), factorial2(x), fib(x), fibonacci(x), Stirling1(n,x), Stirling2(n,x), worpitzky(x), harmonic(x), H(x), zeta(n,x), poly(3,x), hermite(3,x), fallingfactorial(n,m), risingfactorial(n,m), factorialpower(m,n), isprime, prime, nextprime, mod, powermod(m,p,n), cunningham_prime_factors, is_prime, is_prime_power, is_pseudoprime, is_pseudoprime_power, mwrank_initprimes, next_prime, next_prime_power, next_probable_prime, nth_prime, previous_prime, previous_prime_power, prime_divisors, prime_factors, prime_pi, prime_powers, prime_range, prime_to_m_part, primer, primes, primes_first_n, random_prime, twinprime,
special see special function
Statistics max(x), mean(x), median(x), min(x), mode(x), prod(x), std(x), sum(x), total, variance(x)
Trigonometryacos(x), acosh, acot, acoth, acsc, acsch, asec, asech, asin, asinh, atan, atan2, atanh, cos, cosh, cot, coth, csc, csch, sec, sech, sin, sinh, tan, tanh
Utils clone, format, has, hasnot, isinteger, isNaN, isNegative, isnumeric, isPositive, isprime, isfunction, numeric, range,
plot draw, plot(x), parametricplot, implictplot, polarplot, tangentplot, secantplot, funplot, odeplot, complexplot(z),
plot2D plot2D, complex2D, diff2D, integrate2D, function2D, parametric2D, contour2D, semid2D, semiint2D
plot3Dsurface3D, plot3D, parametric3D, implicit3D, spin3D, section3D, contour3D, complex3D, function3D, wireframe3D, math3D, data3D , graph3D

1. ### Math function 数学函数

1. Standard Function 初等函数
2. JavaScript Math function 函数
3. Java Math function 函数

4. Special Function 特殊函数
B Beta beta binomial
Cl Clausen ci chi
dilog digamma doublefactorial
E Ei Ein En eta erf erfc erfi
factorial factorial2 fallingfactorial fresnelC fresnelS
Gamma gamma gaussi
H harmonic
L lerch li loggamma
P Phi pochhammer polygamma polylog psi
si shi smallgamma
mittag
tani
zeta

5. Complex function 复变函数
Basic Functions 基本初等函数, Logarithmic Functions 对数函数, Circular Functions 三角函数, Hyperbolic Functions 双曲函数, Miscellaneous Functions
Gamma Functions 伽马函数, Bessel Functions 贝塞耳函数, Orthogonal Polynomials 正交多项式, Elliptic Integrals 椭圆积分, Elliptic Functions 椭圆函数, Hypergeometric Functions 超几何函数, Zeta Functions

6. Inverse function 反函数
If add prefix "inverse" to function name, it becomes to the inverse function, e.g. inversef(x) is inverse of f(x) for f^-1(x). The inverse of trig function is denoted by prefix "a" onto function name, e.g. asin(x) is inverse of sin(x) for sin^-1(x) , other inverse function is denoted by prefix "inverse" onto function name.

7. polymonial and number theory
bell(x), euler(n,x), factorial(x), factorial2(x), binomial(n,k), bernoulli(n,x), Stirling1(n,x), Stirling2(n,x), worpitzky(x), harmonic(x), zeta(n,x), poly(3,x), hermite(3,x),

2. ### operations

1. Inert function:
If add profix "s" to function name, it becomes to the inert function. Its last letter is "s" for inactivative, e.g. ds( ), integrates( ). A non-integer derivative is inert, e.g. sin^((0.5))(x), to activate by removing its last letter "s" or by clicking the simplify button.

2. to function:
If add prefix "to" to function name, it becomes to the convert function, e.g. adding prefix "to" to sin(x) becomes tosin(x), convert x to sin.
tosin(exp(x))
It is the same as convert( ) function convert to another form. e.g.
convert exp(x) to sin

3. Calculus function 微积分函数
lim limit d diff ds dsolve infint int integral integrate integrates odetest semid semiint series taylor completed

The nth order derivative function 导数
The nthd( ) gives the nth derivative formula. It is symbolic higher derivative. The 0.5th order derivative of sin(x) is denoted by sin(0.5,x) for sin^((0.5))(x). The minus order derivative is intergal, e.g. sin^((-1))(x) is int sin(x), it is different from sin^-1(x), which is inverse of sin(x). A non-integer derivative is inert, e.g. sin^((0.5))(x), to activate by clicking the simplify button.

4. Integral function 积分函数
If add profix "i" or "Integral" to function name, it becomes to the integral function. Its last letter is "i" for integral, e.g. adding profix "i" to tan become tani, tani(x) is integral of tan(x)/x.

5. Discrete math function 离散数学
difference differences Delta sum sums sum0n sum1n sum0inf sum1inf infsum partialsum psum infsum1 partialsum1 psum1 recurrence rsolve tosum

6. Numeric function 数值函数 计算数学
Its first letter is "n" for numeric, e.g. nsum( ) is numeric sum.

3. plotting function 制图函数
plot(x), implicitplot(x=y), parametricplot(t,t), tangentplot(x), secantplot(x), polarplot(x), overlapplot(x), complexplot(z),
制图, 隐函数制图, 参数制图, 切线制图, 割线制图, 极坐标制图,
surface3D(x*y), function3D(x*y),
立体图IE,

The following plot functions did not work on Microsoft IE and QQ browsers. Please use Google Chrome browser.
plot3D(x*y), contour3D(x*y), wireframe3D(x*y), implicit3D(x*y*z), complex3D(x), parametric3D(t,t,t), spin3D(x), section3D(x)
立体图, 立体外形图, 立体线框图, 立体隐函数图, 复数图, 立体参数图,

In order to plot complex function, you should convert it to complex by tocomplex(x) before plot. Use complex(x) to set variable for complex variable in real value, and Use complex(0,x) to set variable for complex variable in imag value. Get real part by re( ) and imag part by im( ). e.g. convert log(x) to re(log(complex(x))).

4. geometry function 几何函数
semicircle(r), circle(r), oval(rx,ry), tangent(x)
5. animation function 动画函数
timer(4), sinwave(4), sunrise(4)

6. Programing function 编程函数
1. "is" function: Its first two letters are "is". e.g. isreal( ), isnumber( ), isconctant( )
2. condition function : if( )
3. repeat function : for( ) While( )
4. block( )

7. Internal Function 内部函数
system function in the Java API.
8. External Function 外部函数
user-defined function in text .txt file, which is auto loaded. You can add your own new function.

Add or define new function of a single character name by f(x)=sin(x); other new function by ff(x_):=sin(x);

## Definition 定义式

Clicking or inputing a function enters it to the Calculator, then click the definition buttion to get its definition.

## Usage 使用方法

convert - gamma - dsolve - solve - rsolve - fsolve - lasolve - nsolve

It is similar to math function in Java and JavaScript. See help and manual.
You can use first function as command and "to" as ",", e.g.

• convert Gamma(x) to x!
then hit the button or the ENTER key in your keybord.

## Function source 源程序

abs activate acos acosh acot acoth acsc acsch arg asec asech asin asinh assume atan atan2 atanh

## Reference

• handbook : chapter 1 function + chapter 12 special function ﻿