Clear[x,y,z,r,R,θ,zprime,d,Current,θhat]
rvector ={x,y,z};
rprime = {r Cos[θ],r Sin[θ],zprime};

j = Current DiracDelta[r-R] DiracDelta[zprime-d] θhat;
Magnitude =Norm[rvector-rprime];

A =
Simplify[
Integrate[
Simplify[
Integrate[
j r^2 Sin[θ]/Magnitude,{r,0,Infinity}],
{R∈Reals,R>0}],
{zprime,-Infinity,Infinity}],
{d∈Reals}]

(* After this point, integration of theta requires elliptical integrals. Once A is solved, Curl(A) gives B *)

μ = 4 Pi 10^-7
k = (4 R r/((r+R)^2+(z-d)^2))^(1/2);

Br = μ Current k/(4 Pi (R r^3)^(1/2))(-(z-d)(EllipticK-(2-k^2)EllipticE/(2(1-k^2))))

Bz = μ Current k/(4 Pi (R r^3)^(1/2))(r(EllipticK+(k^2(r+a)-2r)EllipticE/(2r(1-k^2))))

Current =1
R=1
z=0
d=0
Plot[Br,{r,0,2}]

Plot[EllipticK[m],{m,0,1.1}]