2+3

Plot[Sin[x],{x,0,7}]

(Subscript[J, 0])(x)

Plot[{BesselJ[0,x],BesselJ[1,x]},{x,0,7}]

Plot[{BesselJ[0,x],BesselJ[1,x]},{x,0,7},PlotRange->{{2,7},{0,1}}]

f=Series[Exp[x],{x,0,8}]

g=Normal[f]

g[[5]]

gg=g/.x->-x

(g+gg)/2

Simplify[(g+gg)/2]

Expand[(g+gg)/2]

g

g/.x^n_->0

Power[x,2]

OddQ[90]

h=g/.{x^n_/;OddQ[n]->0}

FullForm[h]

f=.
g=.

f[x_]:=BesselJ[0,x]

g[x_]:=BesselJ[1,x]

pl=Plot[f[x],{x,0,10},PlotStyle->Red]

fsmall=Normal[Series[f[x],{x,0,2}]]

Infinity
∞
∞

flarge=Normal[Series[f[x],{x,∞,1}][[1]]]

pl2=Plot[{fsmall,flarge},{x,0,10},PlotRange->{All,{-0.5,1}},PlotStyle->{{Black,Dashed},{Gray,Dotted}}]

Show[pl,pl2]

Pi
π

Plot[{Cos[x],5x},{x,0,π/2},PlotRange->{All,{0,1.2}}]

FindRoot[Cos[x]==5x,{x,1}]

FindRoot[Cos[x]==x,{x,1}]

Φ[t_?NumericQ]:=x/.FindRoot[Cos[x]==t x,{x,1}]

Φ[5]

Plot[Φ[t],{t,0,5}]

NIntegrate[Φ[t],{t,0,1}]

Ψ[y_]:=NIntegrate[Φ[t],{t,0,y}]

Ψ[1]

Plot[Ψ[y],{y,0,3}]

A={{Cos[x],Sin[x]},{Sin[x],-Cos[x]}}

MatrixForm[A]

A//MatrixForm

A.A

Simplify[A.A]//MatrixForm

Eigenvalues[A]

Eigenvectors[A]

MatrixExp[A t]

FullSimplify[MatrixExp[A t]]

RandomReal[]

n=500;

H=Table[RandomVariate[NormalDistribution[]],{i,1,n},{j,1,n}]

H=H+Transpose[H];

Histogram[Eigenvalues[H],30]

n=20;
H1=Table[RandomVariate[NormalDistribution[]],{i,1,n},{j,1,n}];
H1=H1+Transpose[H1];
H2=Table[RandomVariate[NormalDistribution[]],{i,1,n},{j,1,n}];
H2=H2+Transpose[H2];

ev[a_]:=Eigenvalues[H1 Cos[a]+H2 Sin[a]]

Plot[Sort[ev[a]],{a,0,1},PlotPoints->10]

Plot[Evaluate[Sort[ev[a]]],{a,0,1},PlotPoints->10]

Plot3D[BesselJ[0,Sqrt[x^2+y^2]],{x,-10,10},{y,-10,10}]

D[Sin[Sqrt[x]],x]

D[Sin[Sqrt[x]],x,x]

D[Sin[Sqrt[x]],{x,2}]

Integrate[Exp[-a x],x]

Integrate[Exp[-a x],{x,0,1}]

Integrate[Exp[-a x^6],x]

Integrate[BesselJ[0,x]Exp[-a x^6],x]

Integrate[Exp[-a x],{x,0,∞}]

Integrate[Exp[-a x],{x,0,∞},Assumptions->a>0]

Integrate[Cos[x]/(1+x^2),{x,-∞,∞}]

Integrate[((Cos[x])/(a^2+x^2)),{x,-∞,∞}]

Integrate[((Cos[k x])/(a^2+x^2)),{x,-∞,∞}]

Integrate[((Cos[k x])/(a^2+x^2)),{x,-∞,∞},Assumptions->{k>0,a>0}]

(1/(p^2+a^2))

Integrate[Exp[I p r Cos[θ]],{θ,0,2π}]/(2π)

Integrate[((BesselJ[0,p r])/(p^2+a^2))p,{p,0,∞},Assumptions->{r>0,a>0}]

Plot[BesselK[0,r],{r,0,3}]

Integrate[BesselK[0,x],x]

Integrate[x BesselK[0,x]^2,x]

Integrate[x^3 BesselK[0,x]^2,x]

FullSimplify[%]

R[m_,n_]:=Integrate[((1-Cos[m x+n y])/(2-Cos[x]-Cos[y])),{x,0,2π},{y,0,2π}]/(2π)^2

R[1,0]

Timing[R[1,1]]

Timing[Expand[R[2,0]]]

Timing[Expand[R[3,0]]]

NIntegrate[((1-Cos[100 x])/(2-Cos[x]-Cos[y])),{x,0,2π},{y,0,2π}]/(2π)^2

NIntegrate[((1-Cos[1000 x])/(2-Cos[x]-Cos[y])),{x,0,2π},{y,0,2π}]/(2π)^2

NIntegrate[(1/(2-Cos[x]-Cos[y])),{x,0,2π},{y,0,2π}]/(2π)^2

NIntegrate[(1/(3-Cos[x]-Cos[y]-Cos[z])),{x,0,2π},{y,0,2π},{z,0,2π}]/(2π)^3