JupyterImageResolution = 84;
JupyterOutTextForm = "TeX";

TeX[x_] := ToString[TeXForm[x]]
TeX[x_, y__] := StringJoin[TeX[x], TeX[y]]
TeXRaw[x__, y_] := StringJoin[x, TeX[y]]

MappedBy[x_] := x
MappedBy[x_, F___, G_] := MappedBy[x, F] // G

SetAttributes[TeXEq, HoldFirst]
TeXEq[x_] := TeX[HoldForm[x] == MappedBy[x, ReleaseHold, FullSimplify]]
TeXEq[x_, F__] := TeX[HoldForm[x] == MappedBy[x, ReleaseHold, F]]

Clear[f,x,a,b, JupyterImageResolution]
Integrate[f[x], {x,a,b}]

JupyterImageResolution = 72;
Integrate[f[x], {x,a,b}]

JupyterImageResolution = 84;
Integrate[f[x], {x,a,b}]

Clear[JupyterOutTextForm] (* default *)
Integrate[f[x], {x,a,b}] // TeX

JupyterOutTextForm = "Plain";
Integrate[f[x], {x,a,b}] // TeX

JupyterOutTextForm = "TeX";
Integrate[f[x], {x,a,b}] // TeX

A = {{0,-1},{1,0}};
A // TeX["A=",#]&

MatrixExp[theta A] // TeXEq

JupyterOutTextForm = "Plain";
$Version

JupyterImageResolution = 84
JupyterOutTextForm = "TeX"

Plot3D[Sin[x y], {x, -Pi, Pi}, {y, -Pi, Pi}]

Integrate[Log[Sin[x]], x]

Integrate[Log[Sin[x]], x] // TeXEq

pendulum = {y''[x] + Sin[y[x]] == 0, y[0] == 2, y'[0] == 1};
solpendulum = NDSolve[pendulum, y, {x, -10, 10}]
Plot[y[x] /. solpendulum, {x, -10, 10}]
Plot[Evaluate[{y[x], y'[x], y''[x]} /. solpendulum], {x, -10, 10}]
ParametricPlot[Evaluate[{y[x], y'[x]} /. solpendulum], {x, -10, 10}]

manipulatependulum = Manipulate[
    Module[
        {sol = NDSolve[{y''[x] + Sin[y[x]] == 0, y[0] == p[[1]], y'[0] == p[[2]]}, y, {x, 0, T}]},
        ParametricPlot[Evaluate[{y[x], y'[x]} /. sol], {x, 0, T}, PlotRange -> {{-2Pi, 2Pi}, {-3, 3}}]
    ], {{p, {2, 1}}, Locator}, {{T, 5}, 0, 20}
]

Export["manipulatependulum.nb", manipulatependulum]

we = {D[u[t,x],t,t] == D[u[t,x],x,x], u[0,x]==Exp[-x^2/2], Derivative[1,0][u][0,x] == x Exp[-x^2/2], u[t,-5] == u[t,15]};
we // TeX

solwe = NDSolve[we, u, {t,0,10}, {x,-5,10}]
Plot3D[u[t,x]/.solwe, {t,0,10}, {x,-5,15}, PlotRange->{-0.1,1.1}]
Plot[u[7, x]/.solwe, {x, -5, 15}, PlotRange->{0,1}]

plotswe = Table[Plot[u[t, x]/.solwe, {x, -5, 15}, PlotRange->{0,1}], {t, 0, 10, 0.4}];
Export["wave eq.gif", plotswe, "AnimationRepetitions" -> Infinity]

pdfnormal[s_, x_] := Exp[-x^2/2*s^2]/Sqrt[2*Pi*s^2]
he = {D[u[t,x],t] == D[u[t,x],x,x], u[0,x] == pdfnormal[1/2, x+5] + pdfnormal[2/5, x-5], u[t,-30] == u[t,30]};
he // TeX

solhe = NDSolve[he, u, {t,0,20}, {x,-30,30}]
Table[Plot[u[t, x]/.solhe, {x, -15, 15}, PlotRange->{-0.1,1}], {t, 0, 20, 4}]

plotshe = Table[Plot[u[t, x]/.solhe, {x, -15, 15}, PlotRange->{-0.1,1}], {t, 0, 20, 0.2}];
Export["heat eq.gif", plotshe, "AnimationRepetitions" -> Infinity]

kdv = {D[u[t,x],t] == -D[u[t,x],x,x,x]-6 u[t,x] D[u[t,x], x], u[0,x] == -5 Sin[Pi x/10], u[t,-10] == u[t,10]};
kdv // TeX

solkdv = NDSolveValue[kdv, u, {t,0,1}, {x,-10,10}, MaxStepSize->0.01]
Plot3D[solkdv[t,x], {t,0,0.3}, {x,-10,10}, PlotRange->{-7,15}]
Plot[solkdv[0.15, x], {x, -10, 10}, PlotRange->{-5,15}]

plotskdv = Table[Plot[solkdv[t, x], {x, -10, 10}, PlotRange->{-7,15}], {t, 0, 1, 0.01}];
Export["KdV sine.gif", plotskdv, "AnimationRepetitions" -> Infinity]

??Style

x^2+y^2
Style[x^2+y^2,20]
Style[x^2+y^2,40]
Style[x^2+y^2,60]

L = {{1,2,3}, {2,3,1}, {3,1,2}, {2,2}};
L // TeXEq
SortBy[L, First] // TeXEq
SortBy[L, Last]  // TeXEq
SortBy[L, Total] // TeXEq

A = {{0,1}, {-p^2,0}};
A // TeXEq
MatrixPower[A, 2] // TeX[HoldForm[A^2], "=", #]&
MatrixPower[MatrixPower[A, 2],k] // TeX[HoldForm[A^(2k)], "=", #]&
MatrixPower[MatrixPower[A, 2],k] . A // TeX[HoldForm[A^(2k+1)], "=", #]&

a = ExampleData[{"Audio", "Piano"}];
PitchRecognize[a, AllowedFrequencyRange->{Quantity[200, "Hertz"], Quantity[400, "Hertz"]}] // ListLinePlot

ComplexPlot[(z^2+1)/(z^2-1), {z, -2-2 I, 2+ 2I}, ColorFunction->"CyclicLogAbsArg"]

??Zeta

(* The first nontrivial zero of the Riemann zeta *)
ComplexPlot[Zeta[s], {s, 13.7 I, 1+14.7 I}, ColorFunction->"CyclicLogAbsArg"]

data = Table[a t Exp[2 Pi I t], {a,2,4}, {t,0,1,0.02}];
ComplexListPlot[data, PlotLegends->{"a=2", "a=3", "a=4"}]

zetadata = Table[Zeta[0.6+t I], {t, 0, 100, 0.01}];
ComplexListPlot[zetadata]

Graphics3D[Dodecahedron[]]

Graphics3D[AugmentedPolyhedron[Dodecahedron[]]]

ComplexListPlot[Eigenvalues[RandomVariate[NormalDistribution[], {1000, 1000}]/Sqrt[1000]]]

A = {{2,-1},{-1,2}};
A // TeXEq
ev = Eigensystem[A];
evals = ev[[1]];
evals // TeX["eigen values: ",#]&
evecs = ev[[2]];
Map[MatrixForm, evecs] // TeX["eigen vectors: ",#]&

A = {{2,-1,0},{-1,2,-1},{0,-1,2}};
A // TeXEq
ev = Eigensystem[A];
evals = ev[[1]];
evals // TeX["eigen values: ",#]&
evecs = ev[[2]];
Map[MatrixForm, evecs] // TeX["eigen vectors: ",#]&

A = {{2,-1,0,0},{-1,2,-1,0},{0,-1,2,-1},{0,0,-1,2}};
A // TeXEq
ev = Eigensystem[A];
evals = ev[[1]];
evals // TeX["eigen values: ",#]&
evecs = ev[[2]];
Map[MatrixForm, evecs] // TeX["eigen vectors: ",#]&

A = {{2,-1,0,0,0},{-1,2,-1,0,0},{0,-1,2,-1,0},{0,0,-1,2,-1},{0,0,0,-1,2}};
A // TeXEq
ev = Eigensystem[A];
evals = ev[[1]];
evals // TeX["eigen values: ",#]&
evecs = ev[[2]];
Map[MatrixForm, evecs] // TeX["eigen vectors: ",#]&

A = {
    {2, -1, 0, 0, 0, 0},
    {-1, 2, -1, 0, 0, 0},
    {0, -1, 2, -1, 0, 0},
    {0, 0, -1, 2, -1, 0},
    {0, 0, 0, -1, 2, -1},
    {0, 0, 0, 0, -1, 2}
};
A // TeXEq
ev = Eigensystem[A];
evals = ev[[1]];
N[evals] // TeX["eigen values: ",#]&
evecs = ev[[2]];
Map[MatrixForm, N[evecs]] // TeX["eigen vectors: ",#]&

A = {{27, 48, 81}, {-6, 0, 0}, {1, 0, 3}};
A // TeXEq
Factor[CharacteristicPolynomial[A, x]] // TeX["characteristic polynomial: ", #]&
SJ = JordanDecomposition[A];
S = SJ[[1]];
J = SJ[[2]];
S // TeXEq
J // TeXEq
S . J . Inverse[S] // TeXEq

??MandelbrotSetPlot

MandelbrotSetPlot[]

MandelbrotSetPlot[{-0.52-0.62 I, -0.50-0.60 I}]

??JuliaSetPlot

JuliaSetPlot[0.365 - 0.37 I, PlotLegends -> Automatic, PlotRange->{{-1.2, 1.2}, {-1.5, 1.5}}]

JuliaSetPlot[0.30 - 0.37 I, PlotLegends -> Automatic, PlotRange->{{-1.2, 1.2}, {-1.5, 1.5}}]

JuliaSetPlot[0.4 - 0.37 I, PlotLegends -> Automatic, PlotRange->{{-1.2, 1.2}, {-1.5, 1.5}}]

juliasets = Table[JuliaSetPlot[0.345 + 0.01 t + 0.04 t Abs[t] - 0.37 I, PlotRange->{{-1.2, 1.2}, {-1.5, 1.5}}], {t, -1, 1.5, 0.02}];
Export["JuliaSets.gif", juliasets, "AnimationRepetitions" -> Infinity]

??Integrate

Integrate[Log[Sin[x]], x] // TeXEq

Integrate[Log[Sin[x]], {x, 0, Pi/2}] // TeXEq

Integrate[Log[x]/(1+x^2), x] // TeXEq

Integrate[Log[x]/(1+x^2), {x,0,Infinity}] // TeXEq

Integrate[Log[x]/(1+x^2), {x,1,Infinity}] // TeXEq
Catalan // TeX[#, "=", "Catalan number"]&

??Catalan

??FourierSeries

??FourierTrigSeries

B[n_, x_] := BernoulliB[n, x]
F[n_, x_, N_] := FourierTrigSeries[B[n,x] /. x -> x/(2 Pi) + 1/2, x, N] /. x->2 Pi x - Pi

BB = B[1,x];
BB // TeXRaw["B_1(x)=", #]&
FF = F[1,x,10];
FF // TeX["Fourier series = ", #]&

Plot[{BB, FF}, {x,-0.1,1.1}]

BB = B[2,x];
BB // TeXRaw["B_1(x)=", #]&
FF = F[2,x,10];
FF // TeX["Fourier series = ", #]&

Plot[{BB, FF}, {x,-0.1,1.1}]

??Series

FF = (1+x*h)^(1/h);
FF // TeXRaw["f(h)=",#]&
SS = Series[FF, {h, 0, 3}];
SS // TeXRaw["f(h)=",#]&
NN = Normal[SS];
NN // TeXRaw["f(h)\approx",#]&

Plot[{FF/.x->1, NN/.x->1}, {h,-1,2}]

??PolyGamma

FF = PolyGamma[x];
SS = Series[FF, {x,0,2}];
SS // TeX[FF,"=",#]&
NN = Normal[SS];
NN // StringJoin[TeX[FF], "\approx", TeX[#]]&

Plot[{FF, NN}, {x, 0, 1.5}]

FF = PolyGamma[x];
SS = Series[FF, {x,Infinity,2}];
SS // TeX[FF, "=", #]&
NN = Normal[SS];
NN // StringJoin[TeX[FF], "\approx", TeX[#]]&

Plot[{FF, NN}, {x, 0, 1.5}]

qnum[x_, h_] := (E^(h x)-1)/(E^h-1);
qnum[x,h] // TeXRaw["(x)_h=", #]&
SS = Series[E^h qnum[x,h], {h,0,7}];
SS // TeXRaw["(x)_h=", #]&

sumsofpowers = Table[Factor[k! SeriesCoefficient[SS, k]]/.x->n, {k,1,7}];
sumsofpowers // TeX["sums of powers: ", #]&