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
Number Theory ..... keywords
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
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)=2find a recurrence satisfied by a given sequence
fibonacci
recurrenceAckermann 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
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
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
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...........................................................
