Content
 inverse equation
 polynomial equation
 Algebra_equation
 absolute equation
 Diophantine equation
 congruence equation
 Modulus equation
 Probability_equation
 recurrence_equation
 functional_equation
 difference_equation
 vector_equation
 Inequalities
 system of equations
 2D equations
 2D parametric equations
 3D equations
 3D parametric equations
 4D equations
 differential equation
 fractional differential equation
 integral equation
 fractional integral equation
 differential integral equation
 system of differential equations
 partial differental equation
 fractional partial differental equation
 system of partial differential equations
 test solution
 and
 list
 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
Do exercise and learn from example.
 Fraction `1E21/2`
 Add prefix "big" to number for Big number:
big1234567890123456789  mod operation:
input mod(3,2) for 3 mod 2Complex 复数
math handbook chapter 1.1.2 complexComplex(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)  complex 2D plot:
complex2D(x^x)more are in complex2D(x) show 2 curves of real and imag parts in real domain of real variable x..
 complex animation(z) show animation of complex function in complex domain of complex variable z.
 complexplot(z) show phrase and/or modulus of complex function in complex domain of complex variable z.
 complex 3D plot:
complex3D(pow(x,x))more are in complex function with complex3D(x) in complex domain of complex variable x.
Numerical approximations
 numeric computation end with the equal sign =
sin(pi/4)=  numeric computation with the ≈ button :
sin(pi/4)  numeric computation with the ≈≈ button :
sin(pi/4)  Convert back with numeric computation n(x) :
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=0 by default
tangent( sin(x) )  tangent equation at x=1
tangent( sin(x),x=1 )  tangentplot(x) 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
math handbook chapter 20.5 polynomial  the unit polynomial:
Enter poly(3,x) = poly(3) for the unit polynomial with degree 3: 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)  the zeta( ) polynomial:
zeta(3,x)  simplify(x):
taylor( (x^2  1)/(x1) )  expand(x) polynomial:
expand(hermite(3,x))  topoly(x) convert polynomial to
polys(x) as holder of polynomial coefficients,
convert `x^25*x+6` to poly = topoly( `x^25*x+6` )  simplify polys(x) to polynomial:
simplify( polys(1,5,6,x) )  polyroots(x) is holder of polynomial roots,
topolyroot(x) convert a polynomial to polyroots.
convert (x^21) to polyroot = topolyroot(x^21)  polysolve(x) numerically solve polynomial for multiroots:
polysolve(x^21)  nsolve(x) numerically solve for a single root:
nsolve(x^21)  solve(x) for sybmbloic and numeric roots:
solve(x^21)  construct polynomial from roots, activate polyroots(x) to polynomial, and reduce roots to polynomial equation = 0 by click on the simplify(x) button.
simplify( polyroots(2,3) )Number
When the variable x of polynomial is numnber, it becomes polynomial number, please see Number_Theory section.  expand Trigonometry by expandtrig(x) :
expandtrig( sin(x)^2 )  inverse function :
inverse( sin(x) )  plot a multivalue function by the inverse equation :
inverse( sin(x)=y )  expand trig function :
expandtrig( sin(x)^2 )  expand special function :
expand( gamma(2,x) )  factor(x) :
factor( sin(x)*cos(x) )  complex 2D plot :
complex2D(x^x)more are in complex2D(x) for 2 curves of real and imag parts in complex domain of complex variable x.
 complex animation(z) show animation of complex function in complex domain of complex variable z.
 complexplot(z) show phrase and/or modulus of complex function in complex domain of complex variable z.
 complex 3D plot :
complex3D(pow(x,x)) in complex domain of complex variable x.more are in complex function
special Function
math handbook chapter 12 special Function  click the lim(x) button for Limit at x>0 :
`lim_(x>0) sin(x)/x ` = lim sin(x)/x as x>0 = lim(sin(x)/x)  click the nlim(x) button for numeric limit at x>0 :
nlim(sin(x)/x)  click the limoo(x) button for Limit at x>oo :
`lim _(x>oo) log(x)/x` = lim( log(x)/x as x>inf )
= limoo( log(x)/x )
 one side limit, left or right side :
lim(exp(x),x,0,right)Derivatives
Math Handbook chapter 5 differential calculus  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(x):
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))  Exact answers for integral :
`int (2x+3)^7` dx = int (2x+3)^7  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 1integrator
If integrate(x) cannot do, please try integrator(x) :  integrator(sin(x))
 enter sin(x), then click the ∫ dx button to integrator
fractional integrate
 semi integrate, semiint(x) :
`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 1numeric computation
 numeric computation by click on the "~=" button :
n( `int _0^1` sin(x) dx )numeric integrate
If numeric computation ail, please try numeric integrate nintegrate(x) or nint(x) :
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.  solve x^21=0 step by step
 add ( ) to equation, then add +1:
(x^21=0)+1  add ( ) to equation, then power by 0.5:
(x^2=1)^0.5
inverse an equation
 add ( ) to equation, then add +1:
 inverse an equation to show multivalue curve.
inverse( sin(x)=y )
check its result by clicking the inverse button again.polynomial equation
 polyroots(x) is holder of polynomial roots, topolyroot(x) convert a polynomial to polyroots.
convert (x^21) to polyroot = topolyroot(x^21)  polysolve(x) numerically solve polynomial for multiroots.
polysolve(x^21)  construct polynomial from roots, activate polyroots(x) to polynomial, and reduce roots to polynomial equation = 0 by click on the simplify(x) button.
simplify( polyroots(2,3) )  solve(x) 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(x) numerically solve for a single root.
nsolve(x^21)Algebra Equation f(x)=0
math handbook chapter 3 algebaic Equationsolve(x) also solve other algebra equation, e.g. exp(x) equation,
 Solve nonlinear equations:
solve(exp(x)+exp(x)=4)absolute equation
 solve(x) absolute equation for the unknown x inside the abs(x) function :
input abs(x1)+abs(x2)=3 for
x1+x2=3
click the solve buttonModulus equation
 solve(x) Modulus equation for the unknown x inside the mod(x) function :
input mod(x,2)=1 for
x mod 2 = 1
click the solve button  Enter mod(x^25x+7,2)=1 for
(x^25x+7) mod 2 = 1  Enter mod(x^25x+6,2)=0 for
(x^25x+6) mod 2 = 0congruence equation
a x ≡ b (mod m)math handbook chapter 20.3 congruence
 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^2+3x+2=1*(mod 11)
x^2+3x+2=1 mod(11)2D equations f(x,y) = 0
Diophantine equation f(x,y) = 0
math handbook chapter 20.5 polynomialIt is that number of equation is less than number of the unknown, e.g. one equation with 2 unknowns x and y.
 One 2D equation f(x,y) = 0 for 2 integer solutions x and y
solve( 3x2y2=0, x,y )
solve( x^23x2y2=0, x,y )2D equation f(x,y) = 0 solved graphically
One 2D equation for 2 unknowns x and y, f(x,y) = 0, solved graphically by implicitplot(x)  solve x^2y^2=1 graphically
x^2y^21=02D parametric equations x=f(t), y=g(t)
A system of 2 equations with a parameter t for 2 unknowns x and y, x=f(t), y=g(t), solved graphically :  parametricplot( x=cos(t), y=sin(t) )
 parametric3D( cos(t),sin(t) )
 parametric2D( cos(t),sin(t) )
2D parametric equations x=f(u,v), y=f(u,v)
A system of 3 equations with 2 parameters u and v for 3 unknowns x and y and z, x=f(u,v), y=f(u,v) solved graphically :  parametric3D( cos(u*v),sin(u*v),u*v )
 wireframe3D( cos(x*y),sin(x*y) )
 parametric3D( cos(t),sin(t) )
 2D surface
3D equations
3D parametric equations x=f(t), y=f(t), z=f(t)
A system of 3 equations with a parameter t for 3 unknowns x and y and z, x=f(t), y=f(t), z=f(t), solved graphically :  parametric3D( t,cos(t),sin(t) )
3D parametric equations x=f(u,v), y=f(u,v), z=f(u,v)
A system of 3 equations with 2 parameters u and v for 3 unknowns x and y and z, x=f(u,v), y=f(u,v), z=f(u,v), solved graphically :  parametric3D( u,uv,u*v )
 parametric surface
One 3D equation f(x,y,z) = 0
One equation for 3 unknowns x and y and z, f(x,y,z) = 0, solved graphically :  implicit3D( xyz )
 plot3D( xyz )
4D equations
One 4D equation with 4 variables,  f(x,y,z,t) = 0, solved graphically :
implicit3D( xyzt )  f(x,y,n,t) = 0, solved graphically :
plot2D( xynt )Probability_equation
 solve(x) Probability equation for the unknown k inside the Probability function P(x),
solve( P(x>k)=0.2, k)recurrence_equation
 rsolve(x) recurrence and functional and difference equation for y(x)
y(x+1)+y(x)+x=0
y(x+1)+y(x)+1/x=0  fsolve(x) recurrence and functional and difference equation for f(x)
f(x+1)=f(x)+x
f(x+1)=f(x)+1
functional_equation
 rsolve(x) recurrence and functional and difference equation for y(x)
y(a+b)=y(a)*y(b)
y(a*b)=y(a)+y(b)  fsolve(x) recurrence and functional and difference equation for f(x)
f(a+b)=f(a)*f(b)
f(a*b)=f(a)+f(b)
difference equation
 rsolve(x) recurrence and functional and difference equation for y(x)
y(x+1)y(x)=x
y(x+2)y(x+1)y(x)=0  fsolve(x) recurrence and functional and difference equation for f(x)
f(x+1)f(x)=x
f(x+2) f(x+1)f(x)=0
vector equation
see vectorInequalities
 solve(x) Inequalities for x.
solve( 2*x1>0 )
solve( x^2+3*x+2>0 )system of equations f(x,y)=0, g(x,y)=0
math handbook chanpter 4.3 system of equations  system of 2 equations f(x,y)=0, g(x,y)=0 for 2 unknowns x and y by default if the unknowns omit with the solve() button :
solve( 2x+3y1=0, 3x+2y1=0 )  system of 2 equations f(x,y)=0 and g(x,y)=0 for 2 unknowns x and y by default if the unknowns omit.
On First graph it is solved graphically, where their cross is solution:
( 2x+3y1=0 and 3x+2y1=0 )differential equation
Math handbook chapter 13 differential equation.
ODE(x) and dsolve(x) and lasove(x) solve ordinary differential equation (ODE) for unknown y.Your operation step by step
 solve dy/dx=x/y step by step
 add ( ) to equation, then time by y:
(dy/dx=x/y)*y  add ( ) to equation, then time by dx:
(dy/dx*y=x)*dx  integrate by click the integrate button
int(dy*y=dx*x)  add ( ) to equation, then time by 2:
(1/2y^2=1/2x^2)*2  add ( ) to equation, then power by 0.5:
(y^2=x^2)^0.5
 add ( ) to equation, then time by y:
 Solve linear ordinary differential equations:
y'=x*y+x
y'= 2y
y'y1=0  Solve nonlinear ordinary differential equations:
(y')^22y^24y2=0
dsolve( y' = sin(xy) )
dsolve( y(1,x)=cos(xy) )
dsolve( ds(y)=tan(y) )
dsolve( ds(y)=exp(y) )
dsolve( ds(y)=log(y) )  Solve second order nonlinear ordinary differential equations:
dsolve( ds(y,x,2)=exp(y) )
dsolve( ds(y,x,2)=log(y) )  2000 examples of Ordinary differential equation (ODE)
more examples in bugs
solve graphically
The odeplot(x) 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.  second order ODE
y''=y'yintegral equation
Math handbook chapter 15 integral 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(x) 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^0.5y)/dx^0.5 = y^2*exp(x)`
`(d^0.5y)/dx^0.5 = sin(y)*exp(x)`
`(d^0.5y)/dx^0.5 = exp(y)*exp(x)`
`(d^0.5y)/dx^0.5 = log(y)*exp(x)`
`(d^0.5y)/dx^0.5  a*y^2b*yc = 0`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 and y with a variable t :  linear equations:
dsolve( ds(x,t)=x2y,ds(y,t)=2xy )  nonlinear equations:
dsolve( ds(x,t)=x2y^2,ds(y,t)=2x^2y )  the second order system of 2 equations with 2 unknowns x and y with a variable t :
dsolve( x(2,t)=x,y(2,t)=2xy )  the 0.5th order system of 2 equations with 2 unknowns x and y with a variable t :
dsolve( x(0.5,t)=x,y(0.5,t)=xy )partial differental equation
Math handbook chapter 14 partial differential equation.
PDE(x) and pdsolve(x) solve partial differental equation with two variables t and x for y, 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:
pde `dy/dt = dy/dx2y`  Solve a nonlinear equation:
pde `dy/dt = dy/dx*y^2`
pde `dy/dt = 2dy/dxy^2`
pde `(d^2y)/(dt^2) 2* (d^2y)/(dx^2)y^22x*yx^2=0`partial differential integral equation
 ds(y,t)ints(y,x)yexp(x)=0
pde `(dy)/(dt)int y (dx) yexp(x)=0`fractional partial differental equation
PDE(x) and pdsolve(x) solve fractional partial differental equation.  Solve linear equations:
pde `(d^0.5y)/dt^0.5 = dy/dx2y`  Solve nonlinear equations:
pde `(d^0.5y)/dt^0.5 = 2* (dy)/dx*y^2`
pde `(d^0.5y)/dt^0.5 = 2* (d^0.5y)/dx^0.5y^2`
pde `(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
fractional partial differential integral equation
 ds(y,t)ints(y,x,0.5)exp(x)=0
pde `(dy)/(dt)int y (dx)^0.5 exp(x)=0`  ds(y,t)ints(y,x,0.5)+2yexp(x)=0
pde `(dy)/(dt)int y (dx)^0.5 +2yexp(x)=0`  ds(y,t)ds(y,x)ints(y,x,0.5)+3yexp(x)=0
pde `(dy)/(dt)dy/dyint y (dx)^0.5 +3yexp(x)=0`system of partial differential equations
system of 2 equations with two variables t and x for 2 unknowns y and z:  linear equations:
pde( ds(y,t)ds(y,x)=2z2y,ds(z,t)ds(z,x)=4z4y )  the second order system of 2 equations :
pde( ds(y,t)ds(y,x,2)=2z2y,ds(z,t)ds(z,x,2)=4z4y )  the 0.5th order system of 2 equations :
pde( ds(y,t)ds(y,x,0.5)=2z2y,ds(z,t)ds(z,x,0.5)=4z4y )test solution
test solution for algebaic equation
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
test solution for differential equation to the unknown y by test( solution, eq ) or click the test button :  test( exp(2x), `dy/dx=2y` )
 test( exp(4x), `(d^0.5y)/dx^0.5=2y` )
test solution for recurrence equation to the unknown y
by rtest( solution, eq ) or click the rtest button.test solution for recurrence equation to the unknown f
by ftest( solution, eq ) or click the ftest button.  First graph is in real domain, second graph is in Laplace domain by Lapalce transform
laplace(x)  Input your function, click the laplace button :
laplace(sin(x))inverse laplace transform
 First graph is in Laplace domain , second graph is in real domain by inverse Lapalce transform
inverselaplace(1/x^2)Fourier transform
 First graph is in real domain, second graph is in Fourier domain by Fourier transform
fourier(x)Input your function, click the Fourier button :
 fourier(exp(x))
 sine wave
 Weierstrass function animation
convolution transform
First graph is in real domain, second graph is in convolution domain by convolution transform convolute(x) with x by default: 
Input your function, click the convolute button :
convolute(exp(x))  convoute exp(x) with 1/sqrt(x) :
convolute(exp(x),1/sqrt(x))  Input harmonic(2,x), click the defintion(x) button to show its defintion, check its result by clicking the simplify(x) button,
then click the limoo(x) button for its limit as x>oo.
Difference
 Δ(k^2) = difference(k^2)
Check its result by the sum(x) buttonSummation ∑
Indefinite sum
 ∑ k = sum(k)
 Check its result by the difference(x) 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 :
1+2+ .. +x = `sum _(k=1) ^x k` = sum(k,k,1,x)  Definite sum, sum x from 1 to 5 :
1+2+ .. +5 = ∑(x,x,1,5) = sum(x,x,1,5)
sum(x^k,k,1,5)Definite sum with parameter x as upper limit
sum(k^2, k,1, x)  Check its result by the difference(x) button, and then the expand(x) button.
 convert to sum series definition :
tosum( exp(x) )  expand above sum series by the expand(x) button
:
expand( tosum(exp(x)) )
Indefinite sum
 ∑ k
 sum( x^k/k!,k )

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
 sum from 1 to oo:
Infinite sum of 1/1^2+1/2^2+1/3^2 .. +1/k^2+... = sum( 1/k^2,k,1,oo )  sum from 0 to oo:
Infinite sum of 1/0!+1/1!+1/2! .. +1/k!+... = sum( 1/k!,k,0,oo )  Infinite sum x from 0 to inf :
1/0!+1/1!+1/2!+ .. +1/x! = sum 1/(x!) as x>oo  convert to sum series definition :
tosum( exp(x) ) = toseries( exp(x) )  check its result by clicking the simplify(x) button :
simplify( tosum( exp(x) ))  expand above sum series :
expand( tosum(exp(x)) )  compare to Taylor series with numeric derivative:
taylor( exp(x), x=0, 8)  compare to series with symbolic derivative:
series( exp(x) )  Taylor series expansion as x=0, by default x=0.
taylor( exp(x) as x=0 ) = taylor(exp(x))  series expand not only to taylor series:
series( exp(x) )  but aslo to other series expansion:
series( zeta(2,x) )the fractional order series expansion at x=0 for 5 terms and the 1.5 order
 series( sin(x),x,0,5,1.5 )
Product ∏
 prod(x,x)
 definition of function :
definition( exp(x) )  check its result by clicking the simplify(x) button :
simplify( def(exp(x)) )  convert to series definition :
toseries( exp(x) )  check its result by clicking the simplify(x) button :
simplify( tosum(exp(x)) )  convert to integral definition :
toint( exp(x) )  check its result by clicking the simplify(x) button :
simplify( toint(exp(x)) )  numeric computation end with the equal sign =
sin(pi/4)=  numeric computation with the n(x) ≈≈ button:
n( sin(30 degree) )
n sin(30 degree)  JavaScript numeric calculator with the ≈ button
can calculate numeric, number theory, Probability, Statistics, matrix, solve equation.
more example in JavaScript mathjs
 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 with the funplot ≈ button:
integrate(x=>sin(x),[1,2])more calculus operation in JavaScript calculus
 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(x1,2)=1 for
(x1) mod 2 = 1
Enter mod(x^25x+7,2)=1 for
(x^25x+7) mod 2 = 1
Enter mod(x^25x+6,2)=0 for
(x^25x+6) mod 2 = 0  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 )
solve( x^23x2y2=0, x,y )more example in JavaScript mathjs
 probability of standard normal distribution P(x)
:
P(x<0.8)  standard normal distribution function Phi(x)
:
`Phi(x)`  solve Probability equation for k :
solve(P(x>k)=0.2,k)  combination(4,x)
combination(4,2)more example in JavaScript mathjs
 differentiate :
d(x and x*x)  integrate :
int(x and x*x)list
There are 2 types of list: [1,2,3] is numeric list in JavaScript, and the list(1,2,3) function is symbolic list in mathHand. The symbolic list element can be symbol, formula and function. We can sort list with  mathHand calculator on the = button :
sort(list(2,3,1))  JavaScript calculator on the ≈ button :
[1,2,3].sort()more example in JavaScript mathjs
Statistics 统计
math handbook chapter 16 Statistics  sort(list(x)), add numbers together by total(list()), max(list()), min(list()), size(list()) with mathHand calculator on the = button. e.g.
total(list(1,2,3))  with JavaScript numeric calculator on the ≈ button :
sum([1,2])more example in JavaScript mathjs
Linear Algebra 线性代数 >>
vector
math handbook chapter 8 vectorIt has direction. the position of the element is fix so we cannot sort it. numeric vector is number with direction. the system auto plot the 2dimentional vector. two vector(x) in the same dimention can be operated by +, , *, /, and ^, the result can be checked by its reverse operation.
vector equation
 solve vector(1,2)+x=vector(2,4) is as same as x=vector(2,4)vector(1,2)
 solve 2xvector(2,4)=0 is as same as x=vector(2,4)/2
 solve 2/xvector(2,4)=0 is as same as x=2/vector(2,4)
 solve vector(1,2)*xvector(2,4)=0 is as same as x=vector(2,4)/vector(1,2)
 solve vector(1,2)*x20=0 is as same as x=20/vector(1,2)
 solve vector(2,3)*x+vector(3,2)*y=vector(1,1),x,y
is as same as
solve(1+2*x+3*y=0,1+3*x+2*y=0)
vector calculus
 differentiate vector(x,x) :
d(vector(x,x))  differentiate sin(vector(x,x)) :
d(sin(vector(x,x)))Matrix 复数矩阵
math handbook chapter 4 matrixComplex matrix [[1,2],[3,4]] can be operated by +,,*,/,^,
 with matrix calculator 矩阵计算器
 with JavaScript numeric calculator ≈
more example in JavaScript mathjs
programming 编程 >>
There are many coding :  math coding 数学编程
 HTML + JavaScript coding 网页编程
 cloud computing = web address coding 云计算 = 网址编程 = 网址计算器
Graphics >>
 graphics
 Classification by plot function 按制图函数分类
 Classification by appliaction 按应用分类
Plot 制图 >>
 plane curve 2D
 surface 2D
3D graph 立体图 plot 3D >>
 space curve 3D
 surface 3D
 surface 4D
Drawing 画画 >>
 drawing
Arithmetic 算术 >>
Exact computation
Convert to complex(x)
more are in numeric math
Algebra 代数 >>
tangent
inverse function
Function 函数 >>
Trigonometry 三角函数
Complex Function 复变函数
math handbook chapter 10 Complex Function 复变函数complex2D(x) shows 2 curves of the real and imag parts in real domain x, and complex3D(x) shows complex function in complex domain x, for 20 graphes in one plot.
Calculus 微积分 >>
Limit
math handbook chapter 4.1 limit
differentiate graphically
some functions cannot be differentiated or integrated symbolically, but can be semidifferentiated and integrated graphically in diff2D.Integrals
Math Handbook chapter 6 integral calculusEquation 方程 >>
Your operation step by step
bugs >>
There are over 800 bugs in wolfram software but they are no problem in MathHand.comTransform 转换 >>
Math handbook chapter 11 integral transformlaplace transform
Discrete Math 离散数学 >>
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 polynomial number :
Probability 概率 >>