Handbook
Fraction `1E2-1/2`
Big number: add prefix "big" to number big1234567890123456789
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)

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 )
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 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) )

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) )
inverse

inverse( sin(x) )

plot a multivalue function

inverse( sin(x)=y )

expand

expand( sin(x)^2 )

factor

factor( sin(x)*cos(x) )

`lim _(x->oo) log(x)/x` = lim( log(x)/x as x->inf )

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 activate() or simplify():

activate( 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

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

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)) 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)

Exact answers

`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))

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
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)

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)

mod(x-1,10)=2

3x-2=2*(mod 10)

3x-2=2mod(10)

rsolve() solves recurrence equation to unknown y.

f(x+1)-f(x)=x

solve( 2*x-1>0 )

solve( x^2+3*x+2>0 )

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) )

`int y \ dx = 2y`

`int_0^x (y(t))/sqrt(x-t)` dt = 2y

`(d^0.5y)/dx^0.5=sin(x)`

`d^-0.5/dx^-0.5 y = 2y`

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 )

test solution for differential equation by odetest() or test().

test( exp(2x), `dy/dx=2y` )

test( exp(4x), `(d^0.5y)/dx^0.5=2y` )

2000 examples of Ordinary differential equation (ODE)
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) )
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)) )
numeric solve equation,

nsolve( x^2-5*x+6=0 )

nsolve( x^2-5*x+6 )

numeric integrate, by default x from 0 to 1.

nint( x^2-5*x+6,x,0,1 )

nint x^2-5*x+6 as x from 0 to 1

nint sin(x)

numeric computation,

n( sin(30 degree) ) = n sin(30 degree)
double factorial 6!!
Calculate the 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

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 )
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) )

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) )
semicircle with radius 2, 半园

semicircle(2)

circle with radius 2, 园

circle(2)

oval with x radius 2 and y radius 1, 椭园

oval(2,1)

tangent as x=1, 切线

tangent( sin(x) as x=1 )

by default, x=0, 切线

tangent( sin(x) )

plane curve 平面曲线图

Copyright 2018 DrHuang.com

### Arithmetic 算术 >>

#### Exact computation

#### Complex

polar(1,45degree)

polard(1,45)

2cis(45degree)

tocomplex(polar(1,45degree))

convert 1-i to polar = topolar(1-i)

topolard(1-i)

#### Numerical approximations

n(polar(2,45degree))

n( sin(pi/4) )

n( sin(30 degree) )

n( sin(0.5,1) )

n( sin(1)^0.5 )

### Algebra 代数 >>

simplify( (x^2 - 1)/(x-1) )

expand( (x-1)^3 )

Factorization

factor( x^4-1 )

factor( x^2+3*x+2 )

tangent( sin(x),x=1 )

#### Convert

topower( cos(x) )

convert exp(x) to trig

convert sin(x) to exp = toexp( sin(x) )

toexp(Gamma(2,x))

inverse( sin(x) )

**polymonial:**

convert `x^2-5*x+6` to poly = topoly( `x^2-5*x+6` )

simplify( polys(1,-5,6,x) )

convert (x^2-1) to polyroot = topolyroot(x^2-1)

simplify( polyroots(2,3,x) )

### Trigonometry 三角函数 >>

inverse( sin(x) )

inverse( sin(x)=y )

expand( sin(x)^2 )

factor( sin(x)*cos(x) )

### Calculus 微积分 >>

#### 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

`d/dx sin(x)` = d(sin(x))

`d^2/dx^2 sin(x)` = d(sin(x),x,2) = d(sin(x) as x order 2)

activate( sin(0.5,x) )

`d/dx | _(x=1) x^6` = d( x^6 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

`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))

`sin^((0.5))(x)` = `sin^((0.5))(x)` = sin(0.5,x)

`sin^((0.5))(x)` = simplify(sin(0.5,x))

`d^(0.5)/dx^(0.5) d^(0.5)/dx^(0.5) sin(x)` = d(d(sin(x),x,0.5),x,0.5)

`d^(-0.5)/dx^(-0.5) sin(x)` = d(sin(x),x,-0.5)

^{(-1)}( sin

^{(0.5)}(x) ) = inverse(sin(0.5,x))

`d/dx (f(x)*g(x)*h(x))` = d(f(x)*g(x)*h(x))

`d/dx f(x)/g(x)` = d(f(x)/g(x))

`d/dx ((sin(x)* x^2)/(1 + tan(cot(x))))` = d((sin(x)* x^2)/(1 + tan(cot(x))))

`d/dy cot(x*y)` = d(cot(x*y) ,y)

`d/dx (y(x)^2 - 5*sin(x))` = d(y(x)^2 - 5*sin(x))

` d^n/dx^n (sin(x)*exp(x)) ` = nthd(sin(x)*exp(x))

#### Integrals

`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))

`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)

`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)

`int _1^3` (2*x + 1) dx = int(2x+1,x,1,3) = int(2x+1 as x from 1 to 3)

`int _0^(pi/2)` tan(x) dx =int(tan(x),x,0,pi/2)

`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

n( `int _0^1` sin(x) dx ) = nint(sin(x),x,0,1) = nint(sin(x))

`int sin(x)\ dx^0.5` = `d^(-0.5)/dx^(-0.5) sin(x)` = int(sin(x),x,0.5) = semiint(sin(x))

`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( x^2+3*x+2 )

solve( x^2 + 4*x + a )

solve( x^2 + 4*x + 181 )

nsolve( x^3 + 4*x + 181 )

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

solve( x^2-5*x-6 )

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=2*(mod 10)

3x-2=2mod(10)

**functional equation**

rsolve() solves recurrence equation to unknown y.

f(x+1)-f(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

`d^0.5/dx^0.5 y = 2y`

`d^0.5/dx^0.5 y - exp(x-1)/(x+1)*y = 0`

**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`

dsolve( x(0.5,t)=t,y(0.8,t)=x )

test( exp(2x), `dy/dx=2y` )

test( exp(4x), `(d^0.5y)/dx^0.5=2y` )

### Series 级数 >>

tosum( exp(x) )

simplify( tosum( exp(x) ))

expand( tosum(exp(x)) )

taylor( exp(x), x=0, 8)

series( exp(x) )

taylor( exp(x) as x=0 ) = taylor(exp(x))

by default x=0.

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)

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 ) prod(x)
`prod x`

Δ`k^2` = difference(k^2)

#### Summation ∑

∑ k = sum(k)

Δ`sum k` = difference( sum(k) )

1+2+ .. +x = `sum _(k=1) ^x k` = sum(k,k,1,x)

1+2+ .. +5 = ∑(x,x,0,5) = sum(x,x,0,5)

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)

tosum( exp(x) )

expand( tosum(exp(x)) )

∑ k

sum( x^k/k!,k )

sum(x,x,0,5,1)

infsum( x^k/k!,k )

#### Product ∏

### Definition 定义式 >>

definition( exp(x) )

simplify( def(exp(x)) )

tosum( exp(x) )

simplify( tosum(exp(x)) )

toint( exp(x) )

simplify( toint(exp(x)) )

### Numeric math 数值数学 >>

nsolve( x^2-5*x+6=0 )

nsolve( x^2-5*x+6 )

nint( x^2-5*x+6,x,0,1 )

nint x^2-5*x+6 as x from 0 to 1

nint sin(x)

n( sin(30 degree) ) = n sin(30 degree)

### Number Theory 数论 >>

^{nd}prime prime(4)

3x-1=2*(mod 10)

3x-1=2mod( 10)

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 )

### Plot 制图 >>

sin(x)

plot( sin(x) and x^2)

implicitplot( x=sin(y) )

parametricplot( sin(t) and sin(4*t) )

polarplot( 2*sin(4*x) )

tangentplot( sin(x) )

secantplot( sin(x) )

### Geometry 几何 >>

semicircle(2)

circle(2)

oval(2,1)

tangent( sin(x) as x=1 )

tangent( sin(x) )

### Animation 动画

### 3D graph 立体图 plot3d

**See Also**- Mathematical Symbols

- Mathematics Handbook

- Elementary Math

- Higher Mathematics

- Fractional Calculus

- Fractional differential equation

- Function

- Formula Charts

- Math software: mathHandbook.com
- Math (translattion from Chinese)

- Math handbook (translattion from Chinese)
- Example: