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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

• 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
simplify( (x^2 - 1)/(x-1) )

• expand
expand( (x-1)^3 )

• factor
Factorization
factor( x^4-1 )

• factorizing
factor( x^2+3*x+2 )

• tangent at x=1
tangent( sin(x),x=1 )

#### Convert

• 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
inverse( sin(x) )

polymonial:
• topoly convert polymonial to polys() as holder of polymonial coefficients,
convert x^2-5*x+6 to poly = topoly( x^2-5*x+6 )
• activate polys() to polymonial
simplify( polys(1,-5,6,x) )
• topolyroot convert a polymonial to polyroots() as holder of polymonial roots,
convert (x^2-1) to polyroot = topolyroot(x^2-1)
• activate polyroots() to polymonial
simplify( polyroots(2,3,x) )

### Calculus 微积分 >>

#### Limit

• Limit
• lim_(x->0) sin(x)/x  = lim sin(x)/x as x->0 = lim(sin(x)/x)
lim _(x->oo) log(x)/x = lim( log(x)/x as x->inf )

#### Derivatives

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 evaulate() or simplify():
evaulate( 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))
• 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)
• 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,0,1)
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))

• infinite integrate

integrate
• int sin(x) dx = integrate(sin(x))

#### fractional integrate

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

• ### Equation 方程 >>

#### Algebra Equation

• solve equation and inequalities,
solve( x^2+3*x+2 )

• Symbolic roots
solve( x^2 + 4*x + a )

• Complex roots
solve( x^2 + 4*x + 181 )

• numerical root
nsolve( x^3 + 4*x + 181 )

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

• by default, equation = 0 to default unknown x.
solve( x^2-5*x-6 )

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

#### 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-1,10)=2

• congruence equation
3x-2=2*(mod 10)
3x-2=2mod(10)

#### 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

Some differential equations cannot be solved symbolly, but can be solved graphically by a plot function plot2D( ), or click the plot2D button. e.g. dy/dx = sin(x)-cos(y)

#### integral equation

• integral equation
int y \ dx = 2y
int_0^x (y(t))/sqrt(x-t) dt = 2y
int_0^sqrt(x) (y(t^2)) dt = sin(x)

• 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(0.5,t)=t,y(0.8,t)=x )

#### partial differental equation

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

#### test solution

• test solution for differential equation by odetest() or 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)
• ### Series 级数 >>

• convert to sum series definition
tosum( exp(x) )

• check its result by simplify()
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) )

• ### Discrete Math 离散数学 >>

default index variable in discrete math is k.
• Difference
Δ(k^2) = difference(k^2)

#### Summation ∑

• Indefinite sum
∑ k = sum(k)
• Check its result by difference
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)

• sum(2^k, k,0, x)
• cpnvert to sum series definition
tosum( exp(x) )
• expand above sum series
expand( tosum(exp(x)) )
• Indefinite sum
∑ k
sum( x^k/k!,k )
• partial sum of 1+2+ .. + k = ∑ x = partialsum(k)
• Definite sum of 1+2+ .. +5 = ∑ x
sum(x,x,0,5,1)

• Infinite sum of 1/0!+x/1!+ .. +x^k/k! = sum( x^k/k! as k->oo )
infsum( x^k/k!,k )

#### Product ∏

• prod(x,x)

• prod x

### Definition 定义式 >>

• definition of function
definition( exp(x) )
• check its result by simplify()
simplify( def(exp(x)) )
• convert to sum series definition
tosum( exp(x) )
• check its result by simplify()
simplify( tosum(exp(x)) )
• convert to integral definition
toint( exp(x) )
• check its result by simplify()
simplify( toint(exp(x)) )
• ### Number Theory 数论 >>

• 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 >>

Plot 制图
• plot sin(x) to show solution, by moving mouse wheel to zoom
sin(x)
• plot sin(x) and x^2 to show solutions on cross
plot( sin(x) and x^2)
• implicit plot sin(x)=y to show a multivalue function, by moving mouse wheel to zoom
implicitplot( x=sin(y) )
• parametric plot with default pararmter t
parametricplot( sin(t) and sin(4*t) )
• polar plot
polarplot( 2*sin(4*x) )

• Interactive 互动
• 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) )
• ### Geometry 几何 >>

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

• Interactive 互动
• 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) )
• ### 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

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