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
 system of equations
 2D equations
 2D parametric equations
 3D equations
 3D parametric equations
 4D equations
 Inequalities
 differential equation
 ordinary differential equation (ODE)
 fractional differential equation
 integral equation
 fractional integral equation
 differential integral equation
 system of differential equations
 partial differental equation (PDE)
 fractional partial differental equation
 system of partial differential equations
 test solution
 Classification by plotting function 按制图函数分类
 Classification by dimension 按维数分类
 Classification by appliaction 按应用分类
 Classification by objects 按物体分类
 Classification by function 按函数分类
 Classification by equation 按方程分类
 Classification by domain 按领域分类
 Classification by libary 按文库分类
 Classification by platform 按平台分类
 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`
 mod operation:
input mod(3,2) for 3 mod 2big number in Java
 Add prefix "big" to number for Big integer:
big12345678901234567891  Add prefix "big" to number for Big decimal:
big1.2345678901234567891  numeric computation with the ≈ button :
acos(bignumber(1))  numeric computation end with the equal sign = :
acos(bignumber(1))=Java numeric computation
 numeric computation with the ≈≈ button :
sin(pi/4)  Convert back with numeric computation function 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 )  input complex numbers in the complex plane:
1+2i  in small letters of complex(1,2) is complex number:
complex(1,2)  in first upper case letter of Complex(1,2) is object in JavaScript complex:
Complex(1,2)  input complex number in polar(r,theta) coordinates:
polar(1,pi)  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 trig format of r*cis(theta*degree) :
2cis(45degree)  Convert complex a+b*i to polar(r,theta) coordinates in Java:
convert 1i to polar
topolar(1i)
convert back to a+b*i format by to remove last letter s or click the simplify button  Convert complex a+b*i to polar(r,theta*degree) coordinates:
topolard(1i)  Convert complex a+b*i to polar(r,theta) coordinates in JavaScript:
math.Complex(1,2).toPolar()=
math.complex(1,2).toPolar()=convert to vector
complex(1,2) number is special vector, i.e. the 2dimentional vector, so it can be converted to vector.  to Java vector(1,2)
convert 1i to vector
tovector(1i)  to JavaScript vector [1,2]
math.Complex(1,2).toVector()=
math.complex(1,2).toVector()=complex number plot
 in order to auto plot complex number as vector, input complex(1,2) for 12i.
complex function plot
 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 2D plot:
complex2D(x^x)  input imagary i with variable x for auto plot
exp(i*x)more are in complex2D(x) show 2 curves of real and imag parts in real domain of real variable x..
 complex 3D plot:
complex3D(pow(x,x))  in order to auto plot complex function,
convert exp(x) to complex by
convert exp(x) to complex
convert(exp(x) to complex)
tocomplex(exp(x))more are in complex function with complex3D(x) in complex domain of complex variable x.
 computation in JavaScript should to click the numeric button or expression ending with 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 )  find_root(x,10,10) between 10 and 10 for numeric equation:
find_root( x^25*x+6==0 )
find_root( 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
 simplify:
taylor( (x^2  1)/(x1) )  expand:
expand( (x1)^3 )  factorizing:
factor( x^2+3*x+2 )  factor high order polymonial by factor(x)== :
factor( x^31 )==  factorization:
factor( x^41 )==  combine two terms:
combine( log(a)+log(b) )  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)  put imagary i iwth variable x for auto complex 2D plot :
exp(i*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.Complex
 complex  complex function  complex math
 math handbook chapter 10 complex function
 complex animate(z) for phase animation, the independent variable must be z.
 complex plot(z) for phase and/or modulus, the independent variable must be z.
 complex2D(x) for complex 2 curves of real and imag parts, the independent variable must be x.
 complex3D(x) for 3 dimensional graph, the independent variable must be x.
 color WebXR surface of complex function on complex plane
 Riemann surface  Complex Branches  complex coloring
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 limit(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.vector calculus
 differentiate vector(x,x) :
d(vector(x,x))  differentiate sin(vector(x,x)) :
d(sin(vector(x,x)))  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)  find_root(x) numerically find for a single root.
find_root(x^21)Algebra Equation f(x)=0
math handbook chapter 3 algebaic Equationsolve(x) also solve other algebra equation, e.g.
 nonlinear equations:
solve(exp(x)+exp(x)=4)
solve cos(x)+sin(x)=1if solve(x) cannot solve, then click the numeric button to solve numerically.
 solve(cos(x)sin(x)=1)
then click the numeric buttonabsolute 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 :first order equation
 input mod(3x,5)=1 for
3x mod 5 = 1
click the solve button. it is solved by inversemod(3,5)second order equation
 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
a x ≡ b* (mod m) means two remindars in both sides of equation are the same, i.e. congruence, it is the same as the modular equation mod(a*x,m)=mod(b,m). if b=1, the modular equation mod(a*x,m)=1 can be solved by inversemod(a*x,m). 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, which can be solved by solve(a*xm*y=b,x,y).
first order equation:
 3x = 1*(mod 5)
mod(3x,5) = mod(1,5)second order equation:
 x^2+3x+2 = 1*(mod 11)
x^2+3x+2 = 1 mod(11)
x^2+3x+2 = mod(11)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.  first order Inequalities
solve( 2*x1>0 )  second order Inequalities
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 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  system of 2 equations f(x,y)=0, g(x,y)=0 for 2 unknowns x and y with the solve( ) or solver() button :
solve( [2*x+3*y1, 3*x+2*y1],x,y )  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, 3x+2y1, x,y )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 )differential equation
Math handbook chapter 13 differential equation.
There are two types of differential equations: a single variable is ordinary differential equation (ODE) and multivariables is partial differential equation (PDE).ordinary differential equation (ODE)
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:
 linear ordinary differential equations:
dsolve y'=x*y+x
ode y'= 2y
ode y'y1=0  nonlinear ordinary differential equations:
ode (y')^22y^24y2=0
dsolve( y' = sin(xy) )
dsolve( y(1,x)=acos(y)sin(x)x )
dsolve( ds(y)cos(x)=asin(y)x )
dsolve( ds(y)=exp(y)exp(x) )
dsolve( ds(y)exp(x)=log(y)x )
ode `y'exp(y)1/xx=0`
ode `y'sinh(y)+x1/sqrt(1+x^2)=0`
ode `y'tan(y)+x1/(1+x^2)=0`  second order nonlinear ordinary differential equations:
dsolve( ds(y,x,2)=asin(y)sin(x)x )
dsolve( ds(y,x,2)exp(x)=log(y)x )  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
odeplot y''=y'yintegral equation
Math handbook chapter 15 integral equation.indefinite integral equation
indefinite integral equation  linear equation
input ints(y) 2y = exp(x) for
ode `int y` dx  2y = exp(x)  nonlinear equation
input ints(y) 2y^2 = 0 for
ode `int y` dx  2y^2 = 0double integral equation
input ints(y,x,2) for double integral
 linear equation
ode( `int int y` dx y = exp(x) )  nonlinear equation
ode( `int int y` dx *y= exp(x) )definite integral equation
definite integral equation  linear equation
input integrates(y(t)/sqrt(xt),t,0,x) = 2y for
ode `int_0^x (y(t))/sqrt(xt)` dt = 2y  nonlinear equation
input integrates(y(t)/sqrt(xt),t,0,x) = 2y^2 for
ode `int_0^x (y(t))/sqrt(xt)` dt = 2y^2differential integral equation
 input ds(y)ints(y) yexp(x)=0 for
ode `dy/dxint y dx yexp(x)=0`fractional differential equation
dsolve(x) also solves fractional differential equation  linear equations:
ode `d^0.5/dx^0.5 y = 2y`
ode `d^0.5/dx^0.5 y y exp(4x) = 0`
ode `(d^0.5y)/dx^0.5 y=x`
ode `d^0.5/dx^0.5 y y  E_(0.5) (x^0.5) *x = 0`  nonlinear equations:
ode `(d^0.5y)/dx^0.5 = y^2*exp(x)`
ode `(d^0.5y)/dx^0.5 = sin(y)*exp(x)`
ode `(d^0.5y)/dx^0.5 = exp(y)*exp(x)`
ode `(d^0.5y)/dx^0.5 = log(y)*exp(x)`
ode `(d^0.5y)/dx^0.5  y^22y1 = 0`
ode `(d^0.5y)/dx^0.5  log(y)  exp(x) + x=0`
ode `(d^(1/2)y)/dx^(1/2)asin(y)+xsin(x+pi/4)=0`linear fractional integral equation
 ode `d^0.5/dx^0.5 y = 2y`
nonlinear fractional integral equation
 ode `d^0.5/dx^0.5 y = 2y^2`
fractional differential integral equation
 ds(y,x,0.5)ints(y,x,0.5) yexp(x)=0
ode `(d^0.5y)/(dx^0.5)int y (dx)^0.5 yexp(x)=0`complex order differential equation

ode `(d^(1i) y)/dx^(1i)2yexp(x)=0`
variable order differential equation

ode `(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.  linear equation:
pde `dy/dt = dy/dx2y`  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.  linear equations:
pde `(d^0.5y)/dt^0.5 = dy/dx2y`  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)) )  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:
bell(5)  double factorial 6!!
 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(3x,5)=1 for
3x mod 5 = 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 )prime
 is prime number?
isprime(12321)
is_prime(12321)  Calculate the 4^{th} prime
nthprime(4)
nth_prime(4)
prime(4)  next prime greater than 4
nextprime(4)
next_prime(4)  prime in range 4 to 9
prime_range(4,9)  Probability of standard normal distribution P(range(1,1)) between 1 and 1
 All probability of standard normal distribution P(x) between oo and oo:
P(range(oo,oo)) = 1  probability of standard normal distribution
P(x<0.8)  standard normal distribution function Phi(x):
`Phi(x)`  solve Probability equation for x by default :
solve(P(t>x)=0.2)  the binomial coefficient, choice number, combination number,
binomial(4,2) = combination(4,2) = C(4,2)  arrangement number, permutation(4,x) =
P(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 withmathHand calculator on the = button
 two lists added value :
list(2,3)+list(3,4)  two lists added or join together
[1,2]+[3,4]  convolution of two lists
convolution([1,2],[3,4])JavaScript calculation
 sort with JavaScript calculator:
[1,2,3].sort()=  two lists added value by the ending with =
[1,2]+[3,4]=  JavaScript calculation on the ≈ button by input:
[1,2]+[3,4]
click the ≈ buttonmore 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.
There are two types of vector: symbolic vector(a,b) and numeric vector([1,2])
symbolic vector
 vector(1,2)+vector(3,4)
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)))numeric vector
 vector([1,2])+vector([3,4])
Array 数组
math handbook chapter 4 matrixComplex array [[1,2],[3,4]] can be operated by +,,*,/,^,
 with array calculator 数组计算器
 with JavaScript numeric calculator ≈
more example in JavaScript mathjs
Matrix 矩阵
math handbook chapter 4 matrixComplex matrix([[1,2],[3,4]]) can be operated by +,,*,/,\,^,
 with matrix calculator 矩阵计算器
 matrix([[1,2],[3,4]])
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 >>
 Classification by plotting function 按制图函数分类
 Classification by dimension 按维数分类
 Classification by appliaction 按应用分类
 Classification by objects 按物体分类
 Classification by function 按函数分类
 Classification by equation 按方程分类
 Classification by domain 按领域分类
 Classification by libary 按文库分类
 Classification by platform 按平台分类
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
big number in JavaScript
computation in JavaScript should to click the numeric button or expression ending with equal sign =
The default precision for BigNumber is 64 digits, and can be configured with the option precision.
bignumber(1.23456789123456789)=
Roundoff errors
Calculations with BigNumber are much slower than calculations with Number, but they can be executed with an arbitrary precision. By using a higher precision, it is less likely that roundoff errors occur:// roundoff errors with numbers
math.add(0.1, 0.2) // Number, 0.30000000000000004
math.divide(0.3, 0.2) // Number, 1.4999999999999998
// no roundoff errors with BigNumbers :)
math.add(math.bignumber(0.1), math.bignumber(0.2)) // BigNumber, 0.3
math.divide(math.bignumber(0.3), math.bignumber(0.2)) // BigNumber, 1.5
Numerical approximations
There are two types of numeric computation: JavaScript and Java numeric computation:JavaScript numeric computation
more are in numeric math
Complex 复数
math handbook chapter 1.1.2 complexThere are two types of complex number: JavaScript complex and Java complex:
input complex numbers
Convert complex to polar(r,theta) coordinates
convert back to a+b*i format by to remove last letter s or click the simplify button
Numeric math 数值数学 >>
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 >>
if the right hand side of equation is zero 0, it can omit, e.g. f(x) is the same as f(x)=0.
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
Number Theory 数论 >>
When the variable x of polynomial is numnber, it becomes polynomial number :
more example is number theory in function and in JavaScript javascsript math
Probability 概率 >>