June 3, 2010

A summary of useful commands previously considered

ContourPlot[x^2+x*y+y^2==15, {x,-5,5},{y,-5,5}, Axes->True, Frame->False, AxesLabel->Automatic]

ContourPlot3D[-x^2-y^2+z^2==1,{x,-5,5},{y,-5,5},{z,-5,5}, AxesLabel->Automatic, ViewPoint->{7.3`,-2.4`,2.`}]

N[ Pi^6/945 - Sum[1/i^6,{i,1,50}],15]

N[2/Product[(1-1/(4i^2)),{i,1,10000}],15]              (* Slow Convergence to Pi *)

N[Pi,15]

Series[ArcTan[x], {x,0,100}]

f[x_] = Normal[ Series[ArcTan[x], {x,0,100}] ]

N[ 20f[1/7] + 8 f[3/79], 85]

N[Pi,85]

Quit[]

y = 1 + x + Sum[ a[i]*x^i,{i,2,50}] + O[x]^51

eqn=(1-x^2)D[y,{x,2}]-2x*D[y,x]+12y==0

expr = LogicalExpand[eqn]

sol = Solve[expr]

y =  y /. sol[[1]]

g[x_] = Normal[y]

(1-x^2)g''[x]-2x*g'[x]+12g[x] // Expand

{g[0],g'[0]}

Plot[g[x],{x,-1.2,1.2}, AspectRatio->Automatic]