T[r_,ϕ_,m_]:=T2+(((Ta+Tb)/2)-T2)*((Log[R2/r])/(Log[R2/R1]))+((2(Tb-Ta))/Pi)NSum[((Sin[n Pi/2])/n)*(((R1^n)((R2^(2n))-(r^(2n))))/((r^n)((R2^(2n))-(R1^(2n)))))Cos[n ϕ],{n,1,m}];
T1[ϕ_,m_]:=((Ta+Tb)/2)+((2(Tb-Ta))/Pi)NSum[((Sin[n Pi/2])/n) Cos[n ϕ],{n,1,m}];
ρ[x_,y_]=(Sqrt[(x^2)+(y^2)]);
φ[x_,y_]=ArcTan[y/x]+Pi/2;

T2=0;
Ta=200;
Tb=500;
R1=50;
R2=200;
Plot[T1[ϕ,24],{ϕ,0,2Pi}]

ω=RegionSymmetricDifference[Disk[{0,0},R2,{-Pi/2,Pi/2}],Disk[{0,0},R1,{-Pi/2,Pi/2}]];
RegionPlot[ω]

ContourPlot[T[ρ[x,y],φ[x,y],24],{x,y}∈ω,PlotLegends->Automatic,AspectRatio->2,Contours->20,PlotPoints->20,PlotRange->All]

Plot3D[T[ρ[x,y],φ[x,y],24],{x,y}∈ω,PlotLegends->Automatic,AspectRatio->1, PlotPoints->20,PlotRange->All]