Small z behaviour

Clear["`*"]

f  =  
      a^2/(a^2 + (z - b/2)^2)^(3/
                2)  +   a^2/(a^2 + (z + b/2)^2)^(3/
                2)

Nt  :=  6

t  :=  Normal[Series[f, {z, 0, Nt}]]

Do[c[i]  =  Simplify[t[[i]]/t[[1]]], {i, 1, Nt}]

s  =  Sum[c[i], {i, 1, Nt}] /. {b -> a}

s // TeXForm

t[[1]]

r  =  (125/144 ϵ)^(1/4)

r  /.  ϵ ->   .0001

r /.  ϵ ->  0.01

Large z behaviour

f  =  
      a^2/(a^2 + (z - b/2)^2)^(3/
                2)  +   a^2/(a^2 + (z + b/2)^2)^(3/
                2)

Nt  =  10

q  =  Expand[Normal[Series[f, {z, Infinity, Nt}]]]

q[[10]]

qq  =  
      Expand[q/q[[Nt]] /.  z ->  1/x]

Do[d[i]  = Simplify[ Coefficient[ qq, x^i]], {i, 1, 
        Nt - 4}]

Table[d[i], {i, 1, Nt - 4}]

si  = 1 +   Sum[d[i]/z^i, {i, 1, Nt - 4}]

q[[Nt]]

Clear[g]

g[z_] =  a f  /. {b ->  1, a -> 1}

\(hplot\  = \ 
      Plot[g[z]/g[0], {z, \(-1\), 1}, 
        PlotRange -> {{\(-1\), 1}, \ {0, 1.2}}, 
        AxesLabel -> \ {z, " "}, 
        TextStyle -> \ {FontSize -> \ 14}, 
        Ticks -> \ {{\(-1\), \(-0.5\), 0, \ 0.5, 1}, Automatic}, 
        PlotStyle -> \ {Thickness[0.01], \ Hue[0.9]}, 
        PlotLabel -> \ \*""(Subscript[B, z])(z) for a Helmholtz coil""]\)

Export["C:\Documents and Settings\johnson\My \
Documents\Filexfer\Class06S\helm.eps", hplot]