Content
 inverse equation
 polynomial equation
 Algebra_equation
 system of equations
 Diophantine equation
 congruence equation
 Modulus equation
 Probability_equation
 recurrence_equation
 functional_equation
 Inequalities
 differential equation
 fractional differential equation
 system of differential equations
 partial differental equation
 integral equation
 fractional integral equation
 differential integral equation
 test solution
 list()
 vector
 and
 Interactive plot 互动制图
 parametric plot, polar plot
 solve equation graphically
area plot with integral  complex plot
 Geometry 几何
 plane graph 平面图 with plot2D
 function plot with funplot
 differentiate graphically with diff2D
 integrate graphically with integrate2D
 solve ODE graphically with odeplot
 surface in 3D with plot3D
 contour in 3D with contour3D
 wireframe in 3D with wirefram3D
 complex function in 3D with complex3D
 a line in 3D with parametric3D
 a column in 3D with parametric3D
 the 4dimensional object (x,y,z,t) in 3D with implicit3D
 Fraction `1E21/2`
 Big number: add prefix "big" to number
big1234567890123456789  mod operation
input mod(3,2) for 3 mod 2Complex 复数
Complex( 1,2) number is special vector, i.e. the 2dimentional vector, so it can be operated and plotted as vector.  complex numbers in the complex plane
complex(1,2) = 1+2i  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)  in order to auto plot complex number as vector, input complex(1,2) for 12i,
or convert 12i to complex(1,2) by
convert(12i to complex) = tocomplex(12i)  input complex number in polar
tocomplex(polar(1,45degree))
 Convert complex a+b*i to polar(r,theta) coordinates
convert 1i to polar = topolar(1i)
 Convert complex a+b*i to polar(r,theta*degree) coordinates
topolard(1i)  plot complex
Numerical approximations
 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)/(x1) )  expand
expand( (x1)^3 )  factorization
factor( x^41 )  factorizing
factor( x^2+3*x+2 )  tangent equation at x=1
tangent( sin(x),x=1 )
 tangentplot( ) show 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))  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 multivalue, use the inverse equation instead function.inverse equation
 inverse equation to show multivalue if it has.
inverse( sin(x)=y )
check its result by clicking the inverse button again.polynomial
 the unit polynomial
poly(3,x) gives the unit polynomial x^3+x^2+x+1.  Hermite polynomial
hermite(3,x) gives the Hermite polynomial while hermite(3) gives Hermite number.  harmonic polynomial
harmonic(3,1,x) = harmonic(3,x)
harmonic(3,x)  the zeta polynomial
zeta(3,x) is the zeta polynomial.
 expand polynomial
expand(hermite(3,x))  topoly( ) convert polynomial to polys( ) as holder of polynomial coefficients,
convert `x^25*x+6` to poly = topoly( `x^25*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^21) to polyroot = topolyroot(x^21)
 polysolve( ) numerically solve polynomial for multiroots.
polysolve(x^21)  nsolve( ) numerically solve for a single root.
nsolve(x^21)
 solve( ) for sybmbloic and numeric roots.
solve(x^21)
 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) )Number
When the variable x of polynomial is numnber, it becomes polynomial number, please see Number_Theory section.  Trigonometry 三角函数
expand Trigonometry by expandtrig( )  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 )
Derivatives
 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
 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
f^{(1)}( sin^{(0.5)}(x) ) = inverse(sin(0.5,x))  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))  indefinite integrate
`int` sin(x) dx
= integrate(sin(x))
 enter a function sin(x), then click the ∫ button to integrate
`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))integrator
If integrate( ) cannot do, please try integrator(x)  integrator(sin(x))
 enter sin(x), then click the ∫ dx button to integrator
 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)  Definite integration
`int_0^1` sin(x) dx = integrate( sin(x),x,0,1 ) = integrate sin(x) as x from 0 to 1fractional integrate
 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))integrate graphically
some functions cannot be differentiated or integrated symbolically, but can be semidifferentiated and integrated graphically in integrate2D. e.g.  inverse an equation to show multivalue curve.
inverse( sin(x)=y )
check its result by clicking the inverse button again.polynomial equation
 polyroots( ) is holder of polynomial roots, topolyroot( ) convert a polynomial to polyroots.
convert (x^21) to polyroot = topolyroot(x^21)  polysolve( ) numerically solve polynomial for multiroots.
polysolve(x^21)  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( ) for sybmbloic and numeric roots.
solve(x^21)
solve( x^25*x6 )  solve equation and inequalities, by default, equation = 0 for default unknown x if the unknown omit.
solve( x^2+3*x+2 )  Symbolic roots
solve( x^2 + 4*x + a )  Complex roots
solve( x^2 + 4*x + 181 )  solve equation for x.
solve( x^25*x6=0,x )
 numerically root
nsolve( x^3 + 4*x + 181 )  nsolve( ) numerically solve for a single root.
nsolve(x^21)
Algebra Equation
solve( ) also solve other algebra equation, e.g. exp( ) equation,  Solve nonlinear equations:
solve(exp(x)+exp(x)=4)system of equations
 system of 2 equations with 2 unknowns x and y by default if the unknowns omit.
solve( 2x+3y1=0,3x+2y1=0 )Diophantine equation
 it is that number of equation is less than number of the unknown, e.g. one equation with 2 unknowns x and y.
solve( 2x2yx*y2=0, x,y)
congruence equation
By definition of congruence, a x ≡ b (mod m) if a x − b is divisible by m. Hence, a x ≡ b (mod m) if a x − b = m y, for some integer y. Rearranging the equation to the equivalent form of Diophantine equation a x − m y = b.
x^23x2=2*(mod 2)
x^23x2=2mod(2)
Modulus equation
 solve( ) Modulus equation for the unknown x inside the mod( ) function, e.g.
input mod(x,2)=1 for
x mod 2 = 1
click the solve button
Probability_equation
 solve( ) Probability equation for the unknown k inside the Probability function P( ),
solve( P(x>k)=0.2, k)recurrence_equation
 rsolve( ) recurrence equation for f(x)
f(x+1)=f(x)+x
f(x+1)=f(x)+1/x
functional_equation
 rsolve( ) functional equation for f(x)
f(xy)=f(x)/f(y)
f(x*y)=f(x)+f(y)
Inequalities
 solve( ) Inequalities for x.
solve( 2*x1>0 )
solve( x^2+3*x+2>0 )
differential equation
ODE( ) and dsolve( ) and lasove( ) solve ordinary differential equation (ODE) to unknown y.  Solve linear ordinary differential equations:
y'=x*y+x
y'= 2y
y'y1=0  Solve nonlinear ordinary differential equations:
(y')^22y^24y2=0  2000 examples of Ordinary differential equation (ODE)
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. by default it is first order ODE.
y''=y'y for second order ODEintegral equation
indefinite integral equation
 indefinite integral equation
input ints(y) 2y = exp(x) for
`int y` dx  2y = exp(x)definite integral equation
 definite integral equation
input integrates(y(t)/sqrt(xt),t,0,x) = 2y for
`int_0^x (y(t))/sqrt(xt)` dt = 2ydifferential integral equation
input ds(y)ints(y) yexp(x)=0 for
`dy/dxint y dx yexp(x)=0`fractional differential equation
dsolve( ) also solves fractional differential equation  Solve linear equations:
`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)`  Solve nonlinear equations:
`d^0.5/dx^0.5 y = y^2`fractional integral equation
 `d^0.5/dx^0.5 y = 2y`
fractional differential integral equation
ds(y,x,0.5)ints(y,x,0.5) yexp(x)=0
`(d^0.5y)/(dx^0.5)int y (dx)^0.5 yexp(x)=0`complex order differential equation

`(d^(1i) y)/dx^(1i)2yexp(x)=0`
variable order differential equation
`(d^sin(x) y)/dx^sin(x)2yexp(x)=0`system of differential equations
 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)=xy )partial differental equation
PDE( ) and pdsolve( ) solve partial differental equation with two variables t and x, then click the plot2D button to plot solution, pull the t slider to change the t value. click the plot3D button for 3D graph.  Solve a linear equation:
`dy/dt = dy/dx2y`  Solve a nonlinear equation:
`dy/dt = dy/dx*y^2``(d^2y)/(dt^2) = 2* (d^2y)/(dx^2)y^22x*yx^2`
fractional partial differental equation
PDE( ) and pdsolve( ) solve fractional partial differental equation.  Solve linear equations:
`(d^0.5y)/dt^0.5 = dy/dx2y`  Solve nonlinear equations:
`(d^0.5y)/dt^0.5 = 2* (d^0.5y)/dx^0.5*y^2``(d^0.5y)/dt^0.5 = 2* (d^0.5y)/dx^0.5y^2`
`(d^1.5y)/(dt^1.5) + (d^1.5y)/(dx^1.5)2y^24x*y2x^2 =0`
 More examples are in Analytical Solution of Fractional Differential Equations
test solution
test solution for algebaic equation to the unknown x by test(solution,eq,x) or click the test( ) button.
 test(1,x^21=0,x)
test( 1, x^25*x6 )test solution for differential equation to the unknown y by test( ) or click the test( ) button.
 test( exp(2x), `dy/dx=2y` )
 test( exp(4x), `(d^0.5y)/dx^0.5=2y` )
 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.
Difference
Δ(k^2) = difference(k^2)  Check its result by the sum( ) button
Summation ∑
Indefinite sum
∑ k = sum(k)  Check its result by the difference( ) button
Δ sum(k) = difference( sum(k) )  In order to auto plot, the index variable should be x,
`sum_x x` = sum(x,x)definite sum
 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)) )  partial sum of 1+2+ .. + k = ∑ k = partialsum(k)

Definite sum of 1+2+ .. +5 = ∑ k
partial sum with parameter upper limit x
sum(1/k^2,k,1,x)infinite sum
 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 )  convert to sum series definition
tosum( exp(x) ) = toseries( 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) )Product ∏
 prod(x,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^25*x+6=0 )
nsolve( x^25*x+6 )  numeric integrate, by default x from 0 to 1.
nint( x^25*x+6,x,0,1 )
nint x^25*x+6 as x from 0 to 1
nint sin(x) 
numeric computation,
n( sin(30 degree) ) = n sin(30 degree)  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(bell(5))  double factorial 6!!
 Calculate the 4^{nd} prime prime(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
3x1=2*(mod 2)
x^23x2=2mod( 2)  modular equation
Enter mod(x1,10)=2 for (x1) mod 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( 3x2y2=0, x,y )
 P( ) is probability of standard normal distribution
P(x<0.8)  Phi( ) is standard normal distribution function
`Phi(x)`  solve Probability equation for k
solve(P(x>k)=0.2,k)  list( )
We can sort list( ), add numbers together with total(list()), max(list()), min(list()).  vector( )
It has direction. the position of the element is fix so we cannot sort it. vector is number with direction. two vector( ) in the same dimention can be operated by +, , *, /, ^. the result can be checked by its reverse operation. the system auto plot the 2dimentional vector.
vector(2,4)/vector(1,2) =2
vector(2,4)=vector(1,2)*2  and
We can plot multi curves with the and. e.g. plot(x and x*x). the position of the element is fix so we cannot sort it.Plot 制图 >>
 plane curve 2D
 surface 2D
3D graph 立体图 plot 3D >>
 space curve 3D
 surface 3D
 surface 4D
Animation 动画 >>
 Classication by plot function 按制图函数分类:
 Classication by appliaction 按应用分类:
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 examplesbugs >>
There are over 500 bugs in another software but no problem in MathHand.com
Arithmetic 算术 >>
Exact computation
Convert to complex( )
Algebra 代数 >>
tangent
inverse function
Function 函数 >>
Function 函数
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, for 20 graphes in one plot.special Function
Calculus 微积分 >>
Limit
differentiate graphically
some functions cannot be differentiated or integrated symbolically, but can be semidifferentiated and integrated graphically in diff2D. e.g.Integrals
Equation 方程 >>
inverse an equation
Discrete Math 离散数学 >>
The default index variable in discrete math is k.
sum( x^k/k!,k )
Definite sum with parameter x as upper limit
sum(k^2, k,0, x)Indefinite sum
∑ ksum( x^k/k!,k )
Series 级数
`prod x`
Definition 定义式 >>
series definition
integral definition
Numeric math 数值数学 >>
Number Theory 数论 >>
When the variable x of polynomial is numnber, it becomes number, e.g.