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# Examples of Fractional Calculus Computer Algebra System 例题

Handbook

## Arithmetic 算术 >>

### Exact computation

• Fraction 1E2-1/2

• Big number: add prefix "big" to number
big1234567890123456789

• ### Complex 复数

• 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)

• ### 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 )
• ## Algebra 代数 >>

• 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

• 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))

• ### inverse function

• 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

• 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(3,2x) 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) )

### Number

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

• 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) )

• #### 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.

## Calculus 微积分 >>

### Limit

• 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
(-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))

• ### Integrals

• 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

#### fractional 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

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))
• ## Equation 方程 >>

### inverse equation

• inverse equation to show multivalue if it has.
inverse( sin(x)=y )
check its result by clicking the inverse button again.

### Algebra Equation

• 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 )

### polynomial equation

• 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

• test solution for equation by test( ) or click the test( ) button.
test( -1, x^2-5*x-6 )

### system of 2 equations

• system of 2 equations with 2 unknowns x and y.
solve( 2x+3y-1=0,x+y-1=0, x,y)

### Diophantine equation

• 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)

• Modulus equation
mod(x-1,2)=1

• congruence equation
3x-2=2*(mod 2)
3x-2=2mod(2)

### functional_equation

• rsolve( ) functional equation
f(x+1)-f(x)=x

### Inequalities

• solve( ) Inequalities.
solve( 2*x-1>0 )
solve( x^2+3*x+2>0 )

### differential equation

• 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) )

#### 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

• 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 )

### partial differental equation

• dsolve( ) solves partial differental equation.
dy/dt = dy/dx-2y

#### fractional partial differental equation

• dsolve( ) solves fractional partial differental equation.
d^0.5/dt^0.5 y = dy/dx-2y

#### test solution

• 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)
• ## 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.

### 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) )
• 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)

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

sum(k^2, k,0, x)
• 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)) )

• #### 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

• 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 )

### Product ∏

• prod(x,x)

• prod x

## Series 级数 >>

• 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 定义式 >>

• definition of function
definition( exp(x) )
• check its result by clicking the simplify( ) button
simplify( def(exp(x)) )
• #### series definition

• convert to series definition
toseries( exp(x) )
• check its result by clicking the simplify( ) button
simplify( tosum(exp(x)) )
• #### integral definition

• convert to integral definition
toint( exp(x) )
• check its result by clicking the simplify( ) button
simplify( toint(exp(x)) )
• ## 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 4nd 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
3x-1=2*(mod 10)
3x-1=2mod( 10)
• modular equation
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 )
• ## Probability 概率 >>

• P( ) is probability of standard normal distribution
P(x<0.8)
• Phi( ) is standard normal distribution function
Phi(x)
• ## 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

• 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

#### dynamic plot

• 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

## Geometry 几何 >>

• semicircle with radius 2, 半园
semicircle(2)
• circle with radius 2, 园
circle(2)
oval(2,1)

• #### Dynamic

• 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) )
• ## 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

• plot2D the 3-dimensional object (x,y,t) in 2D plane, manually change the t value by a slider, or tick the auto t checkbox for auto change.
plot2D(x*y-t-1)

### function plot

• funplot a curve, inverse curve in 2D, and calculate expression at x and y=0.

### diff2D

• diff2D numericallyly differentiate a function on graph.

### integrate2D

• integrate2D numericallyly integrate a function on graph.

### to inte

• tointe convert the integrate( ) to the inte( ) for graph. After integration if some functions cannot be integrated, click the tointe( ) button for integral.

### ode plot

• 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 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.

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

• 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

• the 4-dimensional object (x,y,z,t) in 3D space, manually change the t value by a slider
implicit3D(x*y*z-t)

#### Plot 3D with 3 parameters

• parametric3D(cos(t),sin(t),t)
• parametric3D(sin(x),cos(x),x)

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

• parametric3Dxy(x,y,x*y)
• wireframe3Dxy(x,y,x*y)

more examples
• ## programming 编程 >>

online programming 在线编程
1. plot 函数图
2. rose 玫瑰花
3. check code 验证码
4. calculator 计算器
5. sci calculator 科学计算器
6. color 颜色取色器
7. Chinese calendar 农历日历
8. calendar 日历

more examples

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