Handbook
Fraction `1E2-1/2`

Big number: add prefix "big" to number

big1234567890123456789

input complex number in polar(r,theta*degree) coordinates

polar(1,45degree)

input complex number in polar(r,theta) coordinates for degree by polard(r,degree)

polard(1,45)

input complex number in r*cis(theta*degree) format

2cis(45degree)

Convert to complex

tocomplex(polar(1,45degree))

Convert complex a+b*i to polar(r,theta) coordinates

convert 1-i to polar = topolar(1-i)

Convert complex a+b*i to polar(r,theta*degree) coordinates

topolard(1-i)

Convert back by numeric computation n( )

n(polar(2,45degree))

n( sin(pi/4) )

n( sin(30 degree) )

`sin^((0.5))(1)` is the 0.5 order derivative of sin(x) at x=1

n( sin(0.5,1) )
`sin(1)^(0.5)` is the 0.5 power of sin(x) at x=1

n( sin(1)^0.5 )
simplify

taylor( (x^2 - 1)/(x-1) )

expand

expand( (x-1)^3 )

factorization

factor( x^4-1 )

factorizing

factor( x^2+3*x+2 )

tangent equation at x=1

tangent( sin(x),x=1 )

tangentplot( ) show

tangentplot( sin(x) )

convert to exp

toexp( cos(x) )

convert to trig

convert exp(x) to trig

convert sin(x) to exp(x),

convert sin(x) to exp = toexp( sin(x) )

Convert to exp(x)

toexp(Gamma(2,x))

input sin(x), click the inverse button

inverse( sin(x) )

check its result by clicking the inverse button again.

In order to show multi-value, use the inverse equation instead function. inverse equation to show multivalue if it has.

inverse( sin(x)=y )

check its result by clicking the inverse button again.

the unit polynomial

poly(3,x) gives the unit polynomial x^3+x^2+x+1. Hermite polynomial

hermite(3,2x) gives the Hermite polynomial while hermite(3) gives Hermite number. harmonic polynomial

harmonic(-3,1,x) = harmonic(-3,x)

harmonic(-3,2x)

the zeta polynomial

zeta(-3,x) is the zeta polynomial.

topoly( ) convert polynomial to polys( ) as holder of polynomial coefficients,

convert `x^2-5*x+6` to poly = topoly( `x^2-5*x+6` )

activate polys( ) to polynomial

simplify( polys(1,-5,6,x) )

polyroots( ) is holder of polynomial roots, topolyroot( ) convert a polynomial to polyroots.

convert (x^2-1) to polyroot = topolyroot(x^2-1)

polysolve( ) numerically solve polynomial for multi-roots.

polysolve(x^2-1) nsolve( ) numerically solve for a single root.

nsolve(x^2-1)

solve( ) for sybmbloic and numeric roots.

solve(x^2-1)

construct polynomial from roots. activate polyroots( ) to polynomial, and reduce roots to polynomial equation = 0 by click on the simplify( ) button.

simplify( polyroots(2,3) )

Function 函数
Trigonometry 三角函数

inverse function

inverse( sin(x) )

plot a multivalue function by the inverse equation

inverse( sin(x)=y )

expand

expand( sin(x)^2 )

factor

factor( sin(x)*cos(x) )

click the lim( ) button to Limit at x->0

`lim_(x->0) sin(x)/x ` = lim sin(x)/x as x->0 = lim(sin(x)/x) click the nlim( ) button to numeric limit at x->0
click the limoo( ) button to Limit at x->oo

`lim _(x->oo) log(x)/x` = lim( log(x)/x as x->inf )

Differentiate

`d/dx sin(x)` = d(sin(x))

Second order derivative

`d^2/dx^2 sin(x)` = d(sin(x),x,2) = d(sin(x) as x order 2)

sin(0.5,x) is inert holder of the 0.5 order derivative `sin^((0.5))(x)`, it can be activated by simplify( ):

simplify( sin(0.5,x) )

Derivative as x=1

`d/dx | _(x->1) x^6` = d( x^6 as x->1 )

Second order derivative as x=1

`d^2/dx^2| _(x->1) x^6` = d(x^6 as x->1 order 2) = d(x^6, x->1, 2)

Fractional calculus
semiderivative

`d^(0.5)/dx^(0.5) sin(x)` = d(sin(x),x,0.5) = d( sin(x) as x order 0.5) = semid(sin(x))

input sin(0.5,x) as the 0.5 order derivative of sin(x) for

`sin^((0.5))(x)` = `sin^((0.5))(x)` = sin(0.5,x) simplify sin(0.5,x) as the 0.5 order derivative of sin(x),

`sin^((0.5))(x)` = simplify(sin(0.5,x)) 0.5 order derivative again

`d^(0.5)/dx^(0.5) d^(0.5)/dx^(0.5) sin(x)` = d(d(sin(x),x,0.5),x,0.5) Minus order derivative

`d^(-0.5)/dx^(-0.5) sin(x)` = d(sin(x),x,-0.5)

inverse the 0.5 order derivative of sin(x) function

Derive the product rule

`d/dx (f(x)*g(x)*h(x))` = d(f(x)*g(x)*h(x))

… as well as the quotient rule

`d/dx f(x)/g(x)` = d(f(x)/g(x))

for derivatives

`d/dx ((sin(x)* x^2)/(1 + tan(cot(x))))` = d((sin(x)* x^2)/(1 + tan(cot(x))))

Multiple ways to derive functions

`d/dy cot(x*y)` = d(cot(x*y) ,y)

Implicit derivatives, too

`d/dx (y(x)^2 - 5*sin(x))` = d(y(x)^2 - 5*sin(x))

the nth derivative formula

` d^n/dx^n (sin(x)*exp(x)) ` = nthd(sin(x)*exp(x))

click the ∫ button to integrate above result

`int(cos(x)*e^x+sin(x)*e^x)\ dx` = int(cos(x)*e^x+sin(x)*e^x)

`int tan(x)\ dx` = integrate tan(x) = int(tan(x))

Multiple integrate

`int int (x + y)\ dx dy` = int( int(x+y, x),y)

`int int exp(-x)\ dx dx` = integrate(exp(-x) as x order 2)

Definite integration

`int _1^3` (2*x + 1) dx = int(2x+1,x,1,3) = int(2x+1 as x from 1 to 3)

Improper integral

`int _0^(pi/2)` tan(x) dx =int(tan(x),x,0,pi/2)

Infinite integral

`int _0^oo 1/(x^2 + 1)` dx = int(1/x^2+1),x,0,oo)

indefinite integrate
`int` sin(x) dx
= integrate(sin(x))

Definite integration

`int_0^1` sin(x) dx = integrate( sin(x),x,0,1 ) = integrate sin(x) as x from 0 to 1

semi integrate, semiint( )

`int sin(x) \ dx^(1/2)` = int(sin(x),x,1/2) = int sin(x) as x order 1/2 = semiint(sin(x)) = d(sin(x),x,-1/2)

indefinite semiintegrate

`int sin(x)\ dx^0.5` = `d^(-0.5)/dx^(-0.5) sin(x)` = int(sin(x),x,0.5) = semiint(sin(x))

Definite fractional integration

`int_0^1` sin(x) `(dx)^0.5` = integrate( sin(x),x,0.5,0,1 ) = semiintegrate sin(x) as x from 0 to 1

Exact answers

`int (2x+3)^7` dx = int (2x+3)^7

numeric computation by click on the "~=" button

n( `int _0^1` sin(x) dx ) = nint(sin(x),x,0,1) = nint(sin(x))
inverse equation to show multivalue if it has.

inverse( sin(x)=y )

check its result by clicking the inverse button again.

solve equation and inequalities, by default, equation = 0 for default unknown x.

solve( x^2+3*x+2 )

Symbolic roots

solve( x^2 + 4*x + a )

Complex roots

solve( x^2 + 4*x + 181 )

numerically root

nsolve( x^3 + 4*x + 181 )

solve equation to x.

solve( x^2-5*x-6=0 to x )

polyroots( ) is holder of polynomial roots, topolyroot( ) convert a polynomial to polyroots.

convert (x^2-1) to polyroot = topolyroot(x^2-1)

polysolve( ) numerically solve polynomial for multi-roots.

polysolve(x^2-1) nsolve( ) numerically solve for a single root.

nsolve(x^2-1)

solve( ) for sybmbloic and numeric roots.

solve(x^2-1)

construct polynomial from roots. activate polyroots( ) to polynomial, and reduce roots to polynomial equation = 0 by click on the simplify( ) button.

simplify( polyroots(2,3) )

solve( x^2-5*x-6 )

test solution for equation by test( ) or click the test( ) button.

test( -1, x^2-5*x-6 )

system of 2 equations with 2 unknowns x and y.

solve( 2x+3y-1=0,x+y-1=0, x,y)

number of equation is less than number of the unknown, e.g. one equation with 2 unknowns x and y.

solve( 3x-2y-2=0, x,y)

mod(x-1,2)=1

3x-2=2*(mod 2)

3x-2=2mod(2)

rsolve( )

f(x+1)-f(x)=x

solve( ) Inequalities.

solve( 2*x-1>0 )

solve( x^2+3*x+2>0 )

dsolve( ) or lasove( ) solves differential equation to unknown y.

y'=x*y+x

y'= 2y

y'-y-1=0

(y')^2-2y^2-4y-2=0

dsolve( y' = sin(x-y) )
dsolve( y(1,x)=cos(x-y) )
dsolve( ds(y)=tan(x-y) )

`int y \ dx = 2y`

`int_0^x (y(t))/sqrt(x-t)` dt = 2y

`int_0^x y(t)` dt = sin(x) -y

ds(y)-ints(y) -y-exp(x)=0

`dy/dx-int y dx -y-exp(x)=0`

`d^0.5/dx^0.5 y = 2y`

`d^0.5/dx^0.5 y -y - E_(0.5) (4x^0.5) = 0`

`d^0.5/dx^0.5 y -y -exp(4x) = 0`

`(d^0.5y)/dx^0.5=sin(x)`

`d^-0.5/dx^-0.5 y(x) = 2y`

`(d^0.5y)/(dx^0.5)-int y (dx)^0.5 -y-exp(x)=0`

`(d^cos(x) y)/dx^cos(x)-y-exp(x)=0`

system of 2 equations with 2 unknowns x of the 0.5 order and y of the 0.8 order with a variable t.

dsolve( x(1,t)=x,y(1,t)=x-y )

dsolve( ) solves partial differental equation.

`dy/dt = dy/dx-2y`

dsolve( ) solves fractional partial differental equation.

`d^0.5/dt^0.5 y = dy/dx-2y`

test solution for differential equation by test( ) or click the test( ) button.

test( exp(2x), `dy/dx=2y` )

test( exp(4x), `(d^0.5y)/dx^0.5=2y` )

2000 examples of Ordinary differential equation (ODE)
convert to sum series definition

tosum( exp(x) )

check its result by clicking the simplify( ) button

simplify( tosum( exp(x) ))

expand above sum series

expand( tosum(exp(x)) )

compare to Taylor series

taylor( exp(x), x=0, 8)
compare to series

series( exp(x) )

Taylor series expansion as x=0,

taylor( exp(x) as x=0 ) = taylor(exp(x))

by default x=0. series expand not only to taylor series,

series( exp(x) )

but aslo to other series expansion,

series( zeta(2,x) )

definition of function

definition( exp(x) )
check its result by clicking the simplify( ) button

simplify( def(exp(x)) )
convert to series definition

toseries( exp(x) )
check its result by clicking the simplify( ) button

simplify( tosum(exp(x)) )
convert to integral definition

toint( exp(x) )
check its result by clicking the simplify( ) button

simplify( toint(exp(x)) )
numeric solve equation,

nsolve( x^2-5*x+6=0 )

nsolve( x^2-5*x+6 )

numeric integrate, by default x from 0 to 1.

nint( x^2-5*x+6,x,0,1 )

nint x^2-5*x+6 as x from 0 to 1

nint sin(x)
numeric computation,

n( sin(30 degree) ) = n sin(30 degree)

input sin(x)-0.5=0, hit the ENTER key or click the plot button, then put mouse on cross between its curve and the x-axis to show the x-axis value for solution ,

sin(x)-0.5=0
plot sin(x) and x^2, put mouse on cross between two curves to show solution.

plot( sin(x) and x^2)

implicit plot sin(x)=y to show a multivalue function,

implicitplot( x=sin(y) )

parametric plot with default pararmter t

parametricplot( sin(t) and sin(4*t) )

polar plot

polarplot( 2*sin(4*x) )

overlap plot by clicking the overlap button
tangent plot, by moving mouse on the curve to show tangent

tangentplot( sin(x) )
secant plot, by moving mouse on the curve to show secant

secantplot( sin(x) )

more example on graph
semicircle with radius 2, 半园

semicircle(2)
circle with radius 2, 园

circle(2)
oval with x radius 2 and y radius 1, 椭园

oval(2,1)

tangent 切线 as x=1

tangent( sin(x) as x=1 )

切线 by default, at x=0

tangent( sin(x) )

secantnt 割线 at x=0

secantplot( sin(x) )

plot2D(sin(x))

plot2D the 3-dimensional object (x,y,t) in 2D plane, manually change the t value by a slider, or tick the

plot2D(x*y-t-1)

funplot a curve, inverse curve in 2D, and calculate expression at x and y=0.
diff2D numericallyly differentiate a function on graph.
integrate2D numericallyly integrate a function on graph.
tointe convert the integrate( ) to the inte( ) for graph. After integration if some functions cannot be integrated, click the tointe( ) button for integral.
odeplot graphically solve Ordinary Differential Equation (ode).
The odeplot( ) can be used to visualize individual functions, First and Second order Ordinary Differential Equation
over the indicated domain. Input the right hand side of Ordinary Differential Equations, y"=f(x,y,z), where z for y', then click the

## Arithmetic 算术 >>

### Exact computation

big1234567890123456789

### Complex 复数

polar(1,45degree)

polard(1,45)

2cis(45degree)

tocomplex(polar(1,45degree))

convert 1-i to polar = topolar(1-i)

topolard(1-i)

### Numerical approximations

n(polar(2,45degree))

n( sin(pi/4) )

n( sin(30 degree) )

n( sin(0.5,1) )

n( sin(1)^0.5 )

## Algebra 代数 >>

taylor( (x^2 - 1)/(x-1) )

expand( (x-1)^3 )

factor( x^4-1 )

factor( x^2+3*x+2 )

### tangent

tangent( sin(x),x=1 )

**dynamic**tangent line when your mouse over the curve.

tangentplot( sin(x) )

### Convert

convert( sin(x) to exp) is the same as toexp(sin(x))toexp( cos(x) )

convert exp(x) to trig

convert sin(x) to exp = toexp( sin(x) )

toexp(Gamma(2,x))

### inverse function

inverse( sin(x) )

check its result by clicking the inverse button again.

In order to show multi-value, use the inverse equation instead function.

### inverse equation

inverse( sin(x)=y )

check its result by clicking the inverse button again.

### polynomial

poly(3,x) gives the unit polynomial x^3+x^2+x+1.

hermite(3,2x) gives the Hermite polynomial while hermite(3) gives Hermite number.

harmonic(-3,1,x) = harmonic(-3,x)

harmonic(-3,2x)

zeta(-3,x) is the zeta polynomial.

convert `x^2-5*x+6` to poly = topoly( `x^2-5*x+6` )

simplify( polys(1,-5,6,x) )

convert (x^2-1) to polyroot = topolyroot(x^2-1)

polysolve(x^2-1)

nsolve(x^2-1)

solve(x^2-1)

simplify( polyroots(2,3) )

### Number

When the variable x of polynomial is numnber, it becomes polynomial number, please see Number_Theory section.## Function 函数 >>

inverse( sin(x) )

inverse( sin(x)=y )

expand( sin(x)^2 )

factor( sin(x)*cos(x) )

#### Complex Function 复变函数

complex2D( ) shows the real and imag curves in real domain x, and complex3D( ) shows complex function in complex domain z.#### special Function

## Calculus 微积分 >>

### Limit

`lim_(x->0) sin(x)/x ` = lim sin(x)/x as x->0 = lim(sin(x)/x)

`lim _(x->oo) log(x)/x` = lim( log(x)/x as x->inf )

### Derivatives

`d/dx sin(x)` = d(sin(x))

`d^2/dx^2 sin(x)` = d(sin(x),x,2) = d(sin(x) as x order 2)

simplify( sin(0.5,x) )

`d/dx | _(x->1) x^6` = d( x^6 as x->1 )

`d^2/dx^2| _(x->1) x^6` = d(x^6 as x->1 order 2) = d(x^6, x->1, 2)

#### Fractional calculus

`d^(0.5)/dx^(0.5) sin(x)` = d(sin(x),x,0.5) = d( sin(x) as x order 0.5) = semid(sin(x))

`sin^((0.5))(x)` = `sin^((0.5))(x)` = sin(0.5,x)

`sin^((0.5))(x)` = simplify(sin(0.5,x))

`d^(0.5)/dx^(0.5) d^(0.5)/dx^(0.5) sin(x)` = d(d(sin(x),x,0.5),x,0.5)

`d^(-0.5)/dx^(-0.5) sin(x)` = d(sin(x),x,-0.5)

^{(-1)}( sin

^{(0.5)}(x) ) = inverse(sin(0.5,x))

`d/dx (f(x)*g(x)*h(x))` = d(f(x)*g(x)*h(x))

`d/dx f(x)/g(x)` = d(f(x)/g(x))

`d/dx ((sin(x)* x^2)/(1 + tan(cot(x))))` = d((sin(x)* x^2)/(1 + tan(cot(x))))

`d/dy cot(x*y)` = d(cot(x*y) ,y)

`d/dx (y(x)^2 - 5*sin(x))` = d(y(x)^2 - 5*sin(x))

` d^n/dx^n (sin(x)*exp(x)) ` = nthd(sin(x)*exp(x))

### Integrals

`int(cos(x)*e^x+sin(x)*e^x)\ dx` = int(cos(x)*e^x+sin(x)*e^x)

`int tan(x)\ dx` = integrate tan(x) = int(tan(x))

`int int (x + y)\ dx dy` = int( int(x+y, x),y)

`int int exp(-x)\ dx dx` = integrate(exp(-x) as x order 2)

`int _1^3` (2*x + 1) dx = int(2x+1,x,1,3) = int(2x+1 as x from 1 to 3)

`int _0^(pi/2)` tan(x) dx =int(tan(x),x,0,pi/2)

`int _0^oo 1/(x^2 + 1)` dx = int(1/x^2+1),x,0,oo)

`int_0^1` sin(x) dx = integrate( sin(x),x,0,1 ) = integrate sin(x) as x from 0 to 1

#### fractional integrate

`int sin(x) \ dx^(1/2)` = int(sin(x),x,1/2) = int sin(x) as x order 1/2 = semiint(sin(x)) = d(sin(x),x,-1/2)

`int sin(x)\ dx^0.5` = `d^(-0.5)/dx^(-0.5) sin(x)` = int(sin(x),x,0.5) = semiint(sin(x))

`int_0^1` sin(x) `(dx)^0.5` = integrate( sin(x),x,0.5,0,1 ) = semiintegrate sin(x) as x from 0 to 1

`int (2x+3)^7` dx = int (2x+3)^7

n( `int _0^1` sin(x) dx ) = nint(sin(x),x,0,1) = nint(sin(x))

## Equation 方程 >>

### inverse equation

inverse( sin(x)=y )

check its result by clicking the inverse button again.

### Algebra Equation

solve( x^2+3*x+2 )

solve( x^2 + 4*x + a )

solve( x^2 + 4*x + 181 )

nsolve( x^3 + 4*x + 181 )

solve( x^2-5*x-6=0 to x )

### polynomial equation

convert (x^2-1) to polyroot = topolyroot(x^2-1)

polysolve(x^2-1)

nsolve(x^2-1)

solve(x^2-1)

simplify( polyroots(2,3) )

solve( x^2-5*x-6 )

#### test solution

test( -1, x^2-5*x-6 )

### system of 2 equations

solve( 2x+3y-1=0,x+y-1=0, x,y)

### Diophantine equation

solve( 3x-2y-2=0, x,y)

**Modulus equation**

mod(x-1,2)=1

**congruence equation**

3x-2=2*(mod 2)

3x-2=2mod(2)

### functional_equation

**functional equation**

f(x+1)-f(x)=x

### Inequalities

solve( 2*x-1>0 )

solve( x^2+3*x+2>0 )

### differential equation

y'=x*y+x

y'= 2y

y'-y-1=0

(y')^2-2y^2-4y-2=0

#### solve graphically

The odeplot( ) can be used to visualize individual functions, First and Second order Ordinary Differential Equation over the indicated domain. Input the right hand side of Ordinary Differential Equations, y"=f(x,y,z), where z for y', then click the checkbox.### integral equation

**indefinite integral equation**

`int y \ dx = 2y`

**definite integral equation**

`int_0^x (y(t))/sqrt(x-t)` dt = 2y

`int_0^x y(t)` dt = sin(x) -y

**differential integral equation**

ds(y)-ints(y) -y-exp(x)=0

`dy/dx-int y dx -y-exp(x)=0`

#### fractional differential equation

dsolve( ) also solves fractional differential equation`d^0.5/dx^0.5 y = 2y`

`d^0.5/dx^0.5 y -y - E_(0.5) (4x^0.5) = 0`

`d^0.5/dx^0.5 y -y -exp(4x) = 0`

`(d^0.5y)/dx^0.5=sin(x)`

#### fractional integral equation

`d^-0.5/dx^-0.5 y(x) = 2y`

#### fractional differential integral equation

ds(y,x,0.5)-ints(y,x,0.5) -y-exp(x)=0`(d^0.5y)/(dx^0.5)-int y (dx)^0.5 -y-exp(x)=0`

#### variable order differential equation

`(d^sin(x) y)/dx^sin(x)-y-exp(x)=0``(d^cos(x) y)/dx^cos(x)-y-exp(x)=0`

### system of equations

dsolve( x(1,t)=x,y(1,t)=x-y )

### partial differental equation

`dy/dt = dy/dx-2y`

#### fractional partial differental equation

`d^0.5/dt^0.5 y = dy/dx-2y`

#### test solution

test( exp(2x), `dy/dx=2y` )

test( exp(4x), `(d^0.5y)/dx^0.5=2y` )

## Discrete Math 离散数学 >>

The default index variable in discrete math is k.

Input harmonic(2,x), click the defintion( ) button to show its defintion, check its result by clicking the simplify( ) button,
then click the limoo( ) button for its limit as x->oo.

Check its result by the sum( ) button

Check its result by the difference( ) button

Δ sum(k) = difference( sum(k) )
Definite sum, Partial sum x from 1 to x, e.g.

1+2+ .. +x = `sum _(k=1) ^x k` = sum(k,k,1,x) Definite sum, sum x from 1 to 5, e.g.

1+2+ .. +5 = ∑(x,x,0,5) = sum(x,x,0,5) Infinite sum x from 0 to inf, e.g.

1/0!+1/1!+1/2!+ .. +1/x! = sum 1/(x!) as x->oo

sum(x^k,k,0,5)

Check its result by the difference( ) button, and then the expand( ) button.

convert to sum series definition

tosum( exp(x) )
expand above sum series by the expand( ) button

expand( tosum(exp(x)) )

sum( x^k/k!,k )
partial sum of 1+2+ .. + k
= ∑ k
= partialsum(k)
Definite sum of 1+2+ .. +5 = ∑ k

if the upper limit go to infinite, it becomes infinite sum

Infinite sum of 1/1^2+1/2^+1/3^2 .. +1/k^2+... = lim(sum( 1/k^2,k,1,x) as x->oo) = sum( 1/k^2,k,1,oo )

Infinite sum of 1/0!+1/1!+1/2! .. +1/k!+... = lim(sum( 1/k!,k,0,x) as x->oo) = sum( 1/k!,k,0,oo )

prod(x,x)

`prod x`

### Difference

Δ(k^2) = difference(k^2)### Summation ∑

#### Indefinite sum

∑ k = sum(k)Δ sum(k) = difference( sum(k) )

1+2+ .. +x = `sum _(k=1) ^x k` = sum(k,k,1,x)

1+2+ .. +5 = ∑(x,x,0,5) = sum(x,x,0,5)

1/0!+1/1!+1/2!+ .. +1/x! = sum 1/(x!) as x->oo

sum(x^k,k,0,5)

#### Definite sum with parameter x as upper limit

sum(k^2, k,0, x)tosum( exp(x) )

expand( tosum(exp(x)) )

#### Indefinite sum

∑ ksum( x^k/k!,k )

#### partial sum with parameter upper limit x

sum(1/k^2,k,1,x)### infinite sum

Infinite sum of 1/1^2+1/2^+1/3^2 .. +1/k^2+... = lim(sum( 1/k^2,k,1,x) as x->oo) = sum( 1/k^2,k,1,oo )

Infinite sum of 1/0!+1/1!+1/2! .. +1/k!+... = lim(sum( 1/k!,k,0,x) as x->oo) = sum( 1/k!,k,0,oo )

### Product ∏

`prod x`

## Series 级数 >>

tosum( exp(x) )

simplify( tosum( exp(x) ))

expand( tosum(exp(x)) )

taylor( exp(x), x=0, 8)

series( exp(x) )

taylor( exp(x) as x=0 ) = taylor(exp(x))

by default x=0.

series( exp(x) )

but aslo to other series expansion,

series( zeta(2,x) )

## Definition 定义式 >>

definition( exp(x) )

simplify( def(exp(x)) )

#### series definition

toseries( exp(x) )

simplify( tosum(exp(x)) )

#### integral definition

toint( exp(x) )

simplify( toint(exp(x)) )

## Numeric math 数值数学 >>

nsolve( x^2-5*x+6=0 )

nsolve( x^2-5*x+6 )

nint( x^2-5*x+6,x,0,1 )

nint x^2-5*x+6 as x from 0 to 1

nint sin(x)

n( sin(30 degree) ) = n sin(30 degree)

## Number Theory 数论 >>

When the variable x of polynomial is numnber, it becomes number, e.g.
poly number

poly(3,2) Hermite number

hermite(3,2) harmonic number

harmonic(-3,2)

harmonic(-3,2,4)

harmonic(1,1,4) = harmonic(1,4) = harmonic(4) Bell number

n(bellNumber(5)) double factorial 6!!
Calculate the 4
is prime number? isprime(12321)
next prime greater than 4 nextprime(4)
binomial number `((4),(2))`
combination number `C_2^4`
harmonic number `H_4`
congruence equation

3x-1=2*(mod 10)

3x-1=2mod( 10) modular equation

mod(x-1,10)=2

number of equation is less than number of the unknown, e.g. one equation with 2 unkowns x and y,

solve( 3x-2y-2=0, x,y )

poly(3,2)

hermite(3,2)

harmonic(-3,2)

harmonic(-3,2,4)

harmonic(1,1,4) = harmonic(1,4) = harmonic(4)

n(bellNumber(5))

^{nd}prime prime(4)

3x-1=2*(mod 10)

3x-1=2mod( 10)

mod(x-1,10)=2

**Diophantine equation**

number of equation is less than number of the unknown, e.g. one equation with 2 unkowns x and y,

solve( 3x-2y-2=0, x,y )

## Plot 制图 >>

#### Interactive 互动

plot 制图 auto plot when hit the ENTER key, move mouse wheel on graph to zoom and move.#### solve equation graphically

sin(x)-0.5=0

plot( sin(x) and x^2)

implicitplot( x=sin(y) )

parametricplot( sin(t) and sin(4*t) )

polarplot( 2*sin(4*x) )

#### dynamic plot

tangentplot( sin(x) )

secantplot( sin(x) )

more example on graph

## Geometry 几何 >>

semicircle(2)

circle(2)

oval(2,1)

#### Dynamic

tangent( sin(x) as x=1 )

切线 by default, at x=0

tangent( sin(x) )

secantplot( sin(x) )

## plane graph 平面图 plot 2D >>

### plot2D

plot2D a curve, derivative and integral curve in 2D and polar, and calculate expression at x and y=0.plot2D(sin(x))

#### the 3-dimensional object (x,y,t) in 2D plane

**auto t**checkbox for auto change.

plot2D(x*y-t-1)

### function plot

### diff2D

### integrate2D

### to inte

### ode plot

**y'=**checkbox for first order differential_equation, or click the

**y"=**checkbox for second order differential_equation.

## 3D graph 立体图 plot 3D >>

zoom by mouse wheel, and animation by clicking the spin checkbox.
plot3D surface in 3D space plot3DIE(sin(x))
plot3D surface in 3D plot3D(sin(x))
plot3D contour in 3D contour3D(x*y)
plot3D wireframe in 3D wireframe3D(x*y)
complex function in 3D complex3D(sqrt(x))
a line in 3D space parametric3D(x*x)
a column in 3D space parametric3D(x*x+y*y-1)
implicit3D graphically solve the 3-dimensional equation

implicit3D(x*y*z-1)

the 4-dimensional object (x,y,z,t) in 3D space, manually change the t value by a slider

implicit3D(x*y*z-t)

parametric3D(cos(t),sin(t),t)
parametric3D(sin(x),cos(x),x)
parametric3Dxy(x,y,x*y)
wireframe3Dxy(x,y,x*y)

more examples

#### 3D plot with 1 parameter and 1,2,3 variables

implicit3D(x*y*z-1)

#### the 4-dimensional object (x,y,z,t) in 3D space

implicit3D(x*y*z-t)

#### Plot 3D with 3 parameters

#### Plot 3D with 3 parameters and 2 variables x and y

more examples

## programming 编程 >>

online programming 在线编程

- plot 函数图
- rose 玫瑰花
- check code 验证码
- calculator 计算器
- sci calculator 科学计算器
- color 颜色取色器
- Chinese calendar 农历日历
- calendar 日历

more examples

### Animation 动画

**See Also**

- Mathematical Symbols

- Mathematics Handbook

- Elementary Math

- Higher Mathematics

- Fractional Calculus

- Fractional differential equation

- Function

- Formula Charts

- Math software: mathHandbook.com
- Math (translattion from Chinese)

- Math handbook (translattion from Chinese)
- Example: