Wolfram|Alpha introduces a fundamentally new way to get knowledge and answers—

not by searching the web, but by doing dynamic computations based on a vast collection

of built-in data, algorithms, and methods.

**PART 4 : Lets look at the mathematical searches and plotting**

**The following is not intended to be a handbook,**

but only a listing of search examples

but only a listing of search examples

**Number Theory ..... keywords**

Prime Numbers

Prime Numbers

compute a prime factorization

factor 70560

specify a prime by its position in the sequence 2, 3, 5, ...

1,000,000th prime

generate a list of primes

primes <= 100

**Divisors**

compute the divisors of an integer

divisors 3600

compute a greatest common divisor

gcd 24, 36, 48, 60

Diophantine Equations

solve a Diophantine equation

solve 3x+4y=5 over the integers

**Digit Sums**

sum the digits of an integer

add up the digits of 2567345

compute the digit sum in another base

sum of digits of 29225 in base 2

**Special Numbers**

generate Pascal's triangle

7 rows of Pascal's triangle

compute a polygonal number

5th hexagonal number

**Number Theoretic Functions**

get information about a number-theoretic function

Euler phi

do computations with number-theoretic functions

phi(110)

Continued Fractions

Continued Fractions

find the continued fraction representation of a number

continued fraction sqrt(2)

find continued fraction representations of a function

continued fraction tan x

**Algebraic Numbers**

identify algebraic integers and units

Is 1/(1+sqrt(5)) an algebraic integer?

find the minimal polynomial of an algebraic number

minimal polynomial sqrt(2)+sqrt(3)

Number Theory examples are here.............................................................

**Discrete Mathematics ..... keywords**

**Combinatorics**

compute binomial coefficients (combinations)

30 choose 18

compute a Frobenius number

Frobenius number {4, 7, 12}

find partitions of an integer

integer partitions of 10

**Graph Theory**

compute properties of a named graph

Pappus graph

compare several graphs

Petersen graph, icosahedral graph

analyze a graph specified by adjacency rules

1->2, 2->3, 3->1, 3->4, 4->1

**Point Lattices**

compute properties of a lattice

face-centered cubic

compute properties of a root lattice

A4 root lattice

compare several lattices

simple cubic, simple hexagonal

**Sequences**

compute a possible formula and continuation for a sequence

1, 4, 9, 16, 25, ...

sum an incompletely specified infinite series

1/2 + 1/4 + 1/8 + 1/16 + ...

**Recurrences**

solve a recurrence

g(n+1)=n^2+g

specify initial values

f=f(n-1)+f(n-2), f(1)=1, f(2)=2

find a recurrence satisfied by a given sequence

fibonacci recurrence

**Ackermann Function**

evaluate the Ackermann function

ackermann[2,3]

what is ackermann 4,1?

ack(2,ack(2,1))

Discrete Mathematics examples are here............................................................

**Applied Mathematics ..... keywords**

**Optimization**

minimize or maximize a function

maximize x(1-x)e^x

minimize or maximize a function of several variables

maximize 5 + 3x - 4y - x^2 + x y - y^2

minimize or maximize a function subject to a constraint

maximize e^x sin y on x^2+y^2=1

Numerical Integration

Numerical Integration

numerically integrate functions that cannot be integrated symbolically

integrate sin(cos x) from x=0 to 1

integrate sqrt( (1+x^2)/(1+x^4) ) dx, x=0..1

**Dynamical Systems**

analyze the logistic map

logistic map r=3.569935

analyze the Lotka-Volterra predator-prey model

predator-prey model

**Control Systems**

analyze a transfer function model

transfer function (s^2-3)/(-s^3-s+1)

analyze a state space model

state {{0,1,0},{0,0,1},{1/5,-1,0}}, input {{0},{0},{1}}, output {{-3,0,1}}

Applied Mathematics examples are here............................................................

**Mathematical Functions ..... keywords**

**Domain & Range**

compute the domain of a function

domain of f(x) = x/(x^2-1)

compute the range of a function

range of e^(-x^2)

compute domain and range for functions of several variables

domain and range of z = x^2 + y^2

**Special Functions**

compute properties of a special function

zeta(x)

numerically evaluate a special function

sn(2.1, 0.5)

do computations with special functions

d/dx Si(x)^2

**Number Theoretic Functions**

get information about a number-theoretic function

Euler phi

do computations with number-theoretic functions

phi(110)

Representation Formulas

Representation Formulas

find integral representations for a function

gamma(x) integral representation

sqrt(x) continued fraction

Mathematical Functions examples are here............................................................

**Advanced Mathematics ..... keywords**

**Complex Analysis**

compute properties of a function of a complex variable (use the variable z)

e^z

sqrt(z)

**Vector Analysis**

compute the gradient of a function

grad sin(x^2 y)

compute the divergence of a vector field

div [x^2 sin y, y^2 sin xz, xy sin (cos z)]

calculate alternate forms of a vector analysis expression

curl (curl F)

**Knot Theory**

compute properties of a knot

trefoil knot

specify a knot using Alexander-Briggs notation

8_1 knot

compare several knots

Stevedore knot, figure eight knot

**Integral Transforms**

compute a Fourier transform

Fourier transform exp(-x^2)

compute a Laplace transform

LT e^t sin 2t

Advanced Mathematics examples are here...........................................................

**Famous Math Problems ..... keywords**

Open Conjectures

Open Conjectures

get information about a mathematical conjecture

Riemann Hypothesis

continuum hypothesis

ternary Goldbach conjecture

learn about one of the Smale problems

Smale's fifteenth problem

ask a question about a conjecture

Who formulated the halting problem?

statement of the Erdos-Turan conjecture

**Solved Problems**

find out about a solved problem

transcendence of pi

angle trisection

knapsack problem

learn about a mathematical paradox

Cantor's Paradox

get historical information about a theorem

Fermat's little theorem

Konigsberg theorem

Famous Math Problems examples are here...........................................................