f[x_Integer] /; OddQ[x] := x^2

Clear["Global`*"]

v0=1;

c[n_Integer] /;OddQ[n]=4 (v0/(n π)) ;
c[n_] =0;
Table[c[n],{n,-10,10,1}]

term[n_,x_,y_,a_]:=c[n] Sin[((n π x)/a)] Exp[((-n  π y)/a) ]

term[3,x,y,1]

f[x_,y_,n_,a_:1]:=Sum[term[m,x,y,a],{m,1,n}]

f[x,y,5]

f[0.5,0.3,5]

Plot3D[f[x,y,50] //Evaluate,{x,0,1},{y,0,3},PlotRange->All]

Lorain Corson: p. 230

Clear["Global`*"]

e0=1;
a=1;

pot= -e0 r Cos[θ] + ((e0 (a^3)Cos[θ])/(r^2))

tx={x,y} -> CoordinateTransform["Polar"->"Cartesian",{r,θ}] //Thread

tx={r,θ} -> CoordinateTransform["Cartesian" -> "Polar",{x,y}] //Thread

tmp= pot  /.tx

p1=Plot3D[ tmp,{x,-2,2},{y,-2,2}]

Show[p1,p2]

?Circle

?PolarPlot

circle3D[centre_:{0,0,0},radius_:1,normal_:{0,0,1},angle_:{0,2 Pi}]:=Composition[Line,Map[RotationTransform[{{0,0,1},normal},centre],#]&,Map[Append[#,Last@centre]&,#]&,Append[DeleteDuplicates[Most@#],Last@#]&,Level[#,{-2}]&,MeshPrimitives[#,1]&,DiscretizeRegion,If][First@Differences@angle≥2 Pi,Circle[Most@centre,radius],Circle[Most@centre,radius,angle]]

p2=Graphics3D[{Thickness[0.03],Red,circle3D[]}]