Συμβολική επεξεργασία μαθηματικων παραστάσεων

Clear["`*"]

p1=(x-1)(x+2)(x-3)(x+4)

pe1=Expand[p1]

Factor[pe1]

p2=(2x-1)(4x+2)(-6x-3)(2x+4)

p1/p2

p3=(y^2-y+9)*p1

Expand[p3]

Collect[%,y]

p1

p1[[1]]

Length[%]

Length[p1]

pe1

Length[pe1]

PolynomialGCD[p1,p2]

p1

p1//Expand

Exponent[p1,x]

?Exponent

p2//Expand

Coefficient[p2,x^5]

?Coefficient

p4=p1/p2

p4//Simplify

Cancel[p4]

Apart[p4]

Together[p4]

Numerator[p4]

p1//Expand

CoefficientList[p1+x^8,x]

p3

CoefficientList[p3,y]

p1

sol=Solve[p1==0,x]

p1/.x->3

?ReplaceAll

?Replace

?ReplaceRepeated

Replace[b x+3b x^2,{x->b,(x^2)->x}]

ReplaceAll[b x+3b x^2,{x->b,(x^2)->x}]

ReplaceRepeated[b x+3b x^2,{x->b,(x^2)->x}]

a x+3 x^2 -(2/3) x^3/. c_*x^n_.->c/(n+1)!*y^(n+1)

{p1//Expand,sol}

p1/.sol

x/.sol

x1=sol[[1,1,2]]

p5=Sin[2x]^3-Cos[4x]^5

Expand[p5]

tp5=Expand[p5,Trig->True]

Factor[tp5]

p5

Factor[tp5,Trig->True]

%//Simplify

%//FullSimplify

p6=Sin[a+I x]^3- Cos[b+I x]^5

ComplexExpand[p6]

ComplexExpand[p6,{x}]

?TrigExpand

TrigExpand[Cos[8x]]

?TrigToExp

TrigToExp[Sin[3x]]

f[x_]:=Cos[2Log[x]-3Exp[1/x]]

Plot[f[x],{x,1,5}]

TreeForm[f[x]]

Depth[f[x]]

f[x]

Head[f[x]]

Part[f[x],0]

f[x][[0]]

Length[f[x]]

Length[f[x][[1]]]

f[x][[1]]

Level[f[x],1]

Table[{i,Level[f[x],{i}]},{i,-1,Depth[f[x]]}]//MatrixForm

Table[{i,Level[f[x],i]},{i,-1,Depth[f[x]]}]//MatrixForm

p4

Integrate[p4,x]

Integrate[pe1 Cos[x] Sin[x],x]

Integrate[pe1/Sin[x],x]

Names["`*"]

p5

D[p5,x]

FullForm[D[g[x],x]]

D[p5,{x,3}]

Dt[p5]

p3

Dt[p3]

Dt[p3,x]

D[p3,x]

Sum[i^7,{i,1,n}]

Sum[Sinh[k x],{k,1,Infinity}]

Sum[Sinh[k x] y^k/k,{k,1,Infinity}]

Product[(1-a)^k/(5+k)^2,{k,1,n}]

Solve[3Cos[x]- Sin[2x]==0,x] (*,InverseFunctions->True]*)

Solve[{Cos[1/2+a+x]==6,Cos[3/7+a-x]==2}, {a, x}]

Solve[(1+3Sqrt[x]-2x^1/3)x^1/2==0]

DSolve[y''[x]-x y[x]+3==0,y[x],x]

DSolve[{y''[x]-x y[x]+3==0,y[1]==1,y'[1]==0},y[x],x]

DSolve[{y'[x]-y[x]==3,y[1]==1},y[x],x]

de1=y'[x]-x^(1/2) y[x]==72 x

sol=DSolve[{de1,y[1]==0},y[x],x]

sol=sol[[1,1,2]]

Limit[sol,x->0]//N

Limit[sol,x->2]//N

Plot[sol,{x,0,2}]

Limit[Log[Exp[Sqrt[Sin[x]/x]]],x->0]

Plot[Log[Exp[Sqrt[Sin[x]/x]]],{x,-3,3}]

Limit[(1+Pi/x)^x, x->Infinity]

%//N

Plot[(1+Pi/x)^x,{x,1,10000}]

Limit[(2x^3+5)/(x^3-x+4),x->3]//N

Plot[(2x^3+5)/(x^3-x+4),{x,0,6}]

{Limit[2/(2^(1/x) + 1),x->0], Limit[2/(2^(1/x) + 1),x->0, 
Direction -> -1], Limit[2/(2^(1/x) + 1),x->0, Direction -> 1]}

Plot[2/(2^(1/x) + 1),{x,-4,4}]

r=(x - 2)/(x^2 + 2 x)

{Limit[r,x->1], Limit[r,x->0, Direction->-1],Limit[r,x->0, Direction->1],
Limit[r,x->-2, Direction->-1],Limit[r,x->-2, Direction->1]}

Plot[r,{x,-3,3}]

Plot[Sin[1/x], {x,-0.1,0.1}]

Limit[Sin[1/x], x->0]

Plot[x Sin[1/x], {x,-0.1,0.1}]

Limit[x Sin[1/x], x->0]

s1=Series[Exp[x],{x,0,10}]

Normal[s1]

??Series

s2=Series[Exp[x]Sin[2x],{x,0,10}]

s1+s2

s1-s2

s1*s2

Normal[s1*s2]

s1/s2

ComposeSeries[s1,s2]

FullForm[s1]