…
►For
it is possible to use the linear
transformations in such a way that the new arguments lie within the unit circle, except when
.
This is because the linear
transformations map the pair
onto itself.
…
…
►
Laplace Transform Inversion
►Numerical inversion of the Laplace
transform (§
1.14(iii))
…
►
Example. The Discrete Fourier Transform
…
►is called a
discrete Fourier transform pair.
►
The Fast Fourier Transform
…
…
►
20.11.1
…
►If both
are positive, then
allows inversion of its arguments as a modular
transformation (compare (
23.15.3) and (
23.15.4)):
►
20.11.2
…
►With the substitutions
,
, with
, we have
…
►As in §
20.11(ii), the
modulus
of elliptic integrals (§
19.2(ii)), Jacobian elliptic functions (§
22.2), and Weierstrass elliptic functions (§
23.6(ii)) can be expanded in
-series via (
20.9.1).
…
…
►
§10.22(v) Hankel Transform
►The
Hankel transform (or
Bessel transform) of a function
is defined as
…
►For collections of Hankel
transforms see
Erdélyi et al. (1954b, Chapter 8) and
Oberhettinger (1972).
►The following two formulas are generalizations of the Hankel
transform.
…This is the
Weber transform.
…
…
►
15.12.8
►with the branch chosen to be continuous and
when
.
…
►
15.12.9
…
►
…
►By combination of the foregoing results of this subsection with the linear
transformations of §
15.8(i) and the connection formulas of §
15.10(ii), similar asymptotic approximations for
can be obtained with
or
,
.
…