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206 5. Some worked examples arising from physical problems

where B is a second arbitrary constant. The complications alluded to earlier are now
evident: In r as which can never accommodate the condi-
tions at infinity, (5.16b), for any choice of B.
Now any solution of equation (5.13) which admits (5.16b) must balance contribu-
tions from each side of the equation; because the difficulties in (5.18) arise as
and this is where these other boundary conditions are to be applied, we write
where as Whether z also needs to be scaled is unclear at this stage;
let us therefore write and then (5.13) becomes

and so, provided that as we must select We choose
and hence obtain the equation

valid far away from the tube. The solution of this equation is to satisfy the boundary
conditions at infinity and also to match to the solution valid for r = O(1) i.e. to (5.18).
We seek a solution of equation (5.19) in the form

where

which has solutions and the modified Bessel functions; but grows
exponentially as (and decays), so we select the solution

where C is an arbitrary constant. Now this modified Bessel function has the properties:

so the conditions at infinity, (5.16b), are satisfied.
Finally, we match the first term valid for (5.20), to the first term valid
for r = O(1), (5.18). The latter gives