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# mathHandbook.com

## Examples of Fractional Calculus Computer Algebra System

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Handbook

### Arithmetic >>

#### Exact computation

• Fraction 1E2-1/2
• Big number: add prefix "big" to number big1234567890123456789

• #### Complex

• Convert complex a+b*i to polar(r,theta) coordinates
convert 3-4i to polar
topolar(3-4i)

• input complex number in polar(r,theta) coordinates
polar(3,45degree)

• Convert to complex
tocomplex(polar(3,45degree))

• input complex number in cis(theta) format
cis(45degree)
• Convert back by numeric computation
n(cis(45degree))

• #### Numerical approximations

• 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 >>

• 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 power
• topower( 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 >>

#### Limit

lim sin(x)/x as x->0
• lim sin(x)/x as x->0
• lim _(x->oo) log(x)/x

• lim( log(x)/x as x->inf )
by default x=infinity,
• lim(log(x)/x)

#### Derivatives

Differentiate
• d/dx sin(x)
• d(sin(x))

Second order derivative

• d^2/dx^2 sin(x)
• d(sin(x),x,2)

sin(0.5,x) is inert holder of the 0.5 order derivative sin^((0.5))(x), it can be activated by activate() or simplify():

• activate( sin(0.5,x) )
• d^(0.5)/dx^(0.5) sin(x)
• d(sin(x),x,0.5)

semiderivative

• d^(0.5)/dx^(0.5) sin(x)
• semid(sin(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=1,2)
• input sin(0.5,x) as the 0.5 order derivative of sin(x) for sin^((0.5))(x)
sin^((0.5))(x)
• simplify sin(0.5,x) as the 0.5 order derivative of sin(x) for sin^((0.5))(x)
simplify sin^((0.5))(x)
• 0.5 order derivative, semiderivative, semid()
d^0.5/dx^0.5 log(x)
• 0.5 order derivative again
d^(0.5)/dx^(0.5) d^(0.5)/dx^(0.5) sin(x)
• Minus order derivative
d^(-0.5)/dx^(-0.5) sin(x)
• inverse the 0.5 order derivative of sin(x) function
(-1)( sin(0.5)(x) )
• Derive the product rule
d/dx (f(x)*g(x)*h(x))
• …as well as the quotient rule
d/dx f(x)/g(x)
• for derivatives
d/dx ((sin(x)* x^2)/(1 + tan(cot(x))))
• Multiple ways to derive functions
d/dy cot(x*y)
• Implicit derivatives, too
d/dx (y(x)^2 - 5*sin(x))
• the nth derivative formula
 d^n/dx^n (sin(x)*exp(x))

• #### Integrals

• click the ∫ button to integrate above result
int(cos(x)*e^x+sin(x)*e^x) dx
• int tan(x) dx
• semi integrate, semiint()
int sin(x) dx^(1/2)
• Multiple integrate int int (x + y) dx dy
• Definite integration int _1^3 (2*x + 1) dx
• Improper integral int _0^(pi/2) tan(x) dx
• Infinite integral int _0^oo 1/(x^2 + 1) dx
• Exact answers int (2x+3)^7 dx
• numeric computation by click on the "~=" button
n( int _0^1 sin(cos(x)) dx )
• infinite integrate integrate(exp(-x) as x->oo)
integrate
• int sin(x) dx
• integrate(sin(x))

semiintegrate

• d^(-0.5)/dx^(-0.5) sin(x)
• semiint(sin(x))

Definite integration

• int_1^2 sin(x) dx
• integrate( sin(x) as x from 1 to 2 )
• integrate sin(x) as x from 0 to 1

• ### Equation >>

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-10y-2=0, x,y)

• Modulus equation
mod(x-1,10)=2

• congruence equation
3x=2*(mod 10)
3x-2=0 (mod 10)
3x-2=(mod 10)

• functional equation
rsolve() solves recurrence equation to unknown y.
y(x+1)-y(x)=x

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

• differential equation
dsolve() solves differential equation to unknown y.
y'=x*y+x
y'= 2y
y'-y-1=0
(y')^2-2y^2-4y-2=0
• dsolve also solves fractional differential equation
d^0.5/dx^0.5 y = 2y
d^0.5/dx^0.5 y - exp(x-1)/(x+1)*y = 0
• dsolve( y' = sin(x-y) )
• dsolve( y(1,x)=cos(x-y) )
• dsolve( ds(y)=tan(x-y) )

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

• fractional differential equation
(d^0.5y)/dx^0.5=sin(x)

• fractional integral equation
d^-0.5/dx^-0.5 y = 2y

• system of 2 equations with 2 unknowns x and y with variable t.
dsolve( x'=t,y'=x )

• test solution for differential equation by odetest() or test().
odetest( exp(2x), dy/dx=2y )
odetest( exp(4x), (d^0.5y)/dx^0.5=2y )

• 2000 examples of Ordinary differential equation (ODE)
• ### Series >>

• convert to sum series definition
tosum( sin(x) )
• check its result by simplify()
simplify( tosum( sin(x) ))
• expand above sum series
expand( tosum(sin(x)) )
• compare to Taylor series
taylor( sin(x), x=0, 8)
• compare to series
series( sin(x) )

• Taylor series expansion as x=0,
• taylor( exp(x) as x=0 )
by default 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) )

• ### Discrete Mathematics >>

default index variable in discrete math is k.
• Difference Δk^2

• #### Summation ∑

• Indefinite sum ∑ k
• Check its result by difference Δsum k
• Definite sum, Partial sum x from 1 to x, e.g. 1+2+ .. +x =
sum _(k=1) ^x k
• Definite sum, sum x from 1 to 5, e.g. 1+2+ .. +5 =
∑(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( sin(x) )
• expand above sum series
expand( tosum(sin(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)

• prod x

### Definition >>

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

• 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=2*(mod 10)
• 3x-2=0 (mod 10), 3x-2=(mod 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-10y-2=0, x,y )
• ### Probability >>

• P() is probability of standard normal distribution
P(x<1)
• Phi() is standard normal distribution function
Phi(x)
• ### Plot >>

• 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) )
• ### Geometry >>

• 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) )
circle(2)
• plane curve
• ### Animation >>

Add time function for animation, time(0) is timer to count from 0 to 60.
• +time(0) moving to left.
sin(x+time(0))
• -time(0) moving to right.
sin(x-time(0))
• sine function run after cosine function.
cos(x-time(0))+1 and sin(x-time(0))-1
• tangent rides on sine function at x=0.
sin(x+time(0)) and cos(time(0))*x+sin(time(0))
• heart moving
polarplot(sin(x-time(0))-1)
• flower grows up.
polarplot(sin(4*x)*time(0)/10)
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