(*************************************)
(* Short introduction to Mathematica *)
(* Karl w Broman, 27 April 2007      *)
(*************************************)

(********** first stuff **********)
5^12
%1 ^ (1/12)
% + a
L = 3
L
L = 3;
20 L
2 m + 3 m
%%
%8


(********** help **********)
?Factor*
?FactorInteger
??FactorInteger

?*Graphic*
?*Plot*

?@


(********** packages **********)
GramSchmidt[{{1,1,0}, {0,2,1}, {1,0,3}}]
Remove[GramSchmidt]
<< LinearAlgebra`Orthogonalization`
GramSchmidt[{{1,1,0}, {0,2,1}, {1,0,3}}]


(********** numbers **********)
5 * 10
5 10
a10
a 10

1/2 + 2/144
1/2 + 2.0/144

Sqrt[27]
Sqrt[27.0]
N[Sqrt[27]]


(********** constants **********)
E
Pi
N[E,25]
N[Pi,100]


(********** algebra **********)
Expand[ (x + 2y + z)^2 ]
Factor[ % ]
Together[ 1/(1+2x) - 2/(2+3x) ]
Apart[ % ]      


(********** solving equations **********)
f = x^3 - 3 x^2 - 17x + 51;
soln = Solve[f == 0, x]
f /. soln
NRoots[ f == 0, x ]


(********** simple example **********)
Vg = a^2 / 2 + d^2 / 4;
Ve = Vg (1-hsq)/hsq;
a = 4d;
hsq = 6/10;
Solve[Ve == 1, d]


(********** solving systems **********)
eqn1 = 2 p1 + 2 p2 == 1;
eqn2 = p1 == (1-r) p1 + p2 / 2;
Solve[ {eqn1, eqn2}, {p1, p2}]


(********** nonlinear example **********)
N[ Solve[ {x^2 == 2y + 2, x == y^2 + 1}, {x,y} ] ]

<< Miscellaneous`RealOnly`
Solve[ {x^2 == 2y + 2, x == y^2 + 1}, {x,y} ]

(********** series **********)
Sum[ Exp[-mu] mu^n / Factorial[n], {n, 0, Infinity} ]
Sum[ n Exp[-mu] mu^n / Factorial[n], {n, 0, Infinity} ]
Sum[ (n - mu)^2 Exp[-mu] mu^n / Factorial[n], {n, 0, Infinity} ]

Sum[ p^k, {k, 0, n} ]
Sum[ p^k, {k, 1, n}]


(********** limits **********)
Limit[ Sin[x]/x, x -> 0 ]
Limit[ 1/x, x -> Infinity ]
Limit[ 1/x, x->0, Direction -> -1 ]
Limit[ 1/x, x->0, Direction -> 1 ]


(********** integrals & derivatives **********)
Integrate[ x^4 Cos[x], x ]
D[%, x]
Simplify[%]

Integrate[ Exp[x], {x, -1, 1} ]
Together[%]

(********** another example **********)
Integrate[ 3 Sqrt[3]/(2 Pi) / (r^2 + r(1-r) + (1-r)^2), r]
% /. r -> 0

Solve[%1 + 1/2 == 0.025, r]