BAUER-FORMULA

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DERIVATION OF THE BAUER FORMULA 
 
 
In 1859 the German mathematician Bauer(Agricola in Latin and Farmer in 
English) published the formula- 
 
 )(cos()()12()cos(exp()exp(
0
θθ nn
n
n Pkrjinikrikz ∑ +==
∞
=
 
 
where jn(kr) are the spherical Bessel functions of the first kind of order n and 
Pn(cos(θ)) the Legendre polynomials of order n. When I first encountered this 
relation in my undergraduate nuclear physics class I had absolutely no idea how this 
relation comes about, but I did learn that it was useful in understanding plane wave 
scattering off of a sphere. With knowledge of the Helmholtz equation discussed 
earlier in our PDE course, it is now a relatively simple matter to derive this identity. 
The derivation follows. 
 
Start with the Helmholtz equation for polar axis-symmetric wave propagation 
expressed in spherical coordinates- 
 
 0sin
sin
11 2
2
2
2 =+∂
∂
∂
∂
+
∂
∂
∂
∂ ψ
θ
ψθ
θθ
ψ k
rr
r
rr
 
 
 
Separate the variables by the substitution ψ=R(r)F(θ) and use the separation 
constant n(n+1). This produces the two ordinary differential equations- 
 
 0)]1()[('2" 22 =+−++ RnnkrrRRr 
and- 
 
 0))(sin1(][sin =++ Fnn
d
dF
d
d θ
θ
θ
θ
 
 
 
The first of these equations is just the spherical Bessel equation and has a solution 
 ( finite at r=0 ) of - 
 
 
kr
krJkrjrR nn
)(
2
)()( 2/1+== π 
 
 
The second equation can be converted into a standard Legendre equation by the 
substitution of x=cos(θ). It reads- 
 
 0)1()1( 2 =++− Fnn
dx
dFx
dx
d
 
 
and has a solution( finite at x=1) of- 
 
 nn
n
nn xdx
d
n
xPxF )1(
2!
1)()( 2 −== 
 
 Expressed in terms of x one has- 
 
 xxPxPxPP ]35[
2
1,]13[
2
1,,1 23
2
210 −=−=== 
 
The total wave function solution to the above Helmholtz equation thus becomes- 
 
 )()(
0
xPkrjC nn
n
n∑=
∞
=
ψ 
 
where the coefficient Cn is to be determined. Let us now assume that ψ represents a 
simple incoming wave of wavenumber k. Expressed in spherical 
coordinates(independent of azimuthal angle φ) it reads – 
 
 === )exp()exp( ikrxikzψ )()(
0
xPkrjC nn
n
n∑
∞
=
 
 
Next using the orthogonality conditions- 
 
 
nmm
r
nnm
x
mn n
kxdkrjkrjand
n
dxxPxP δπδ
12
)2/()()()(
12
2)()(
0
1
1 +
=∫∫
+
=
∞
=
+
−=
 
 
 
one finds that the expansion coefficient Cn becomes- 
 
 dxkrdxPkrjikrxnC
r
nn
x
n )()]()()[exp(
)12(
0
1
1
2
∫ ∫
+
=
∞
=
+
−=π
 
 
 
 
 
Evaluating the Cns we find- 
 
)12(
,3)()]([)2(9,1)()((2
0
2
11
0
2
0
+=
∫ ===∫=
∞
=
∞
=
niCwith
ikrdkrjiCkrdkrjC
n
n
rr
o ππ 
 
An extra benefit following from this evaluation is the interesting identity- 
 
 ∫ =
+
−=
1
1
)(2)exp()(
x
n
n
n krjidxikrxxP 
 
which we have not been able to find in any mathematical tables but one which must 
be correct for all positive integer values of n. Looking at this last integral one can 
say that the right hand side represents the finite Legendre transform of the 
exponential function exp(ikrx). 
 
The above results for Cn now allow us to directly write down the sought after Bauer 
Formula- 
 
 )(cos()()12()exp(
0
θnn
n
n Pkrjinikz ∑ +=
∞
=
 
 
Expanding the first few terms in this result yields- 
 
...]1)(cos3][)cos(
)(
)sin([
2
5cos)sin(31)exp( 22 +−−−+= θθ kr
kr
kr
kr
kr
kriikz 
 
Note that this expansion looks quite different from a straight forward Taylor 
expansion- 
 
...)(cos
!2
)(cos1)cosexp()exp( 2
2
+−+== θθθ krikrikrikz 
 
but should yield the same value for exp(ikz) as one sums an infinite number of terms 
in the given two infinite series. 
 
 
 October 2009