Abstract
We define the hyperbolic form factor of a density distribution as its bilateral Laplace transform, related by duality or analytic continuation to its ordinary form factor. For a sphere it is given by
$$\Phi (x \!= \! kR) =\langle \,\cosh \,\vec k.\vec r\,\rangle = \langle \,\frac{\sinh kr}{kr}\,\rangle $$
Φ
(
x
=
k
R
)
=
⟨
cosh
k
→
.
r
→
⟩
=
⟨
sinh
k
r
kr
⟩
, expanded as
$$\,\sum _0^\infty \frac{x^{2n}}{(2n+1)! }\, \frac{\langle r^{2n}\rangle }{R^{2n}}$$
∑
0
∞
x
2
n
(
2
n
+
1
)
!
⟨
r
2
n
⟩
R
2
n
, and similarly for the form factor
$$ \langle \,\frac{\sin kr}{kr}\,\rangle $$
⟨
sin
k
r
kr
⟩
. It is also obtained from the bilateral Laplace transform of
$$2\pi r\,\rho (|r|)$$
2
π
r
ρ
(
|
r
|
)
, and enters in the determination of the outside Yukawa potential induced by a new charge for a mediator of mass
$$m= k=$$
m
=
k
=
$$1/\lambda $$
1
/
λ
.
$$\Phi (x)$$
Φ
(
x
)
may be expressed as
$$\frac{3}{x^3}\,(x\,\cosh x - \sinh x) \times {{\bar{\rho }}} (x)/\rho _0$$
3
x
3
(
x
cosh
x
-
sinh
x
)
×
ρ
¯
(
x
)
/
ρ
0
, where
$${{\bar{\rho }}}(x)$$
ρ
¯
(
x
)
is an effective density decreasing (for
$$d\rho /dr <0)$$
d
ρ
/
d
r
<
0
)
from the average
$$\rho _0$$
ρ
0
at small
x
, down to the density
$$\rho (R)$$
ρ
(
R
)
near the surface. An inversion formula allows one to recover the density distribution
$$\rho (r)$$
ρ
(
r
)
from an analytic continuation of
$$\Phi (x)$$
Φ
(
x
)
, as
$$\rho (r) =\rho _0\, (2R/3\pi r)\int _0^\infty \Phi (ix) \, \sin (x\frac{r}{R})\ x\,dx\,$$
ρ
(
r
)
=
ρ
0
(
2
R
/
3
π
r
)
∫
0
∞
Φ
(
i
x
)
sin
(
x
r
R
)
x
d
x
.