[2] Sister M.C. Fasenmyer, Bull. Amer. Math. Soc. 53, 806
(1947).
[3] Sister M.C. Fasenmyer, Am. Math. Monthly 56, 14
(1949).
[4] G.N. Watson, A Treatise on the Theory of Bessel Function (Cambridge University Press, Cambridge, 1966).
[5] D.C. Chang and F.S. Barnes, Scientiﬁc Report n. 2
(University of Colorado, Boulder, 1973).
[6] E.F. Kuester, Scientiﬁc Report n .72 (University of Colorado, Boulder, 1982).
 √ 

K0 μ p p + d ept dp =

Γ

⎧
 

e−td/2
d 2
⎪
2
⎪
cosh
t −μ
⎨ 2
2
t − μ2
⎪
⎪
⎩
0

Concluding remarks

t>μ
t < μ,

where Γ is the modiﬁed Bromwich contour [10] and
d ≥ 0. Taking the same limit as above we can get
⎧
1
⎪
⎪

t>μ
⎨ 2
1
t − μ2
K0 (μp) ept dp =
⎪
2πi Γ
⎪
⎩
0
t < μ,
which can be interpreted as the inverse Laplace transform of the second kind modiﬁed Bessel function.
[7] W.A. Rodrigues Jr., Equações Diferenciais Parciais da
Fı́sica-Matemática (book in Portuguese, in preparation, 2008).
[8] N. Bleistein, Mathematical Methods for Wave Phenomena (Academic Press, Inc., New York, 1984).
[9] A. Sezginer and W.C. Chew, IEEE Trans. Anten. Propag. AP-32, 527 (1984).
[10] E. Capelas de Oliveira e W.A. Rodrigues Jr., Variáveis
Complexas e Aplicações (Editora Livraria da Fı́sica,
São Paulo, 2006), in Portuguese.
[11] E. Capelas de Oliveira, Funções Especiais com Aplicações (Editora Livraria da Fı́sica, São Paulo, 2005),
in Portuguese.
[12] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals,
Series, and Products (Academic Press, Inc., New York,
1980).

### Language ###
English

### Assessment ###
The text contains mathematical formulas and their transformations, particularly dealing with Bessel functions and inverse Laplace transforms.

### Query ###
What is the inverse Laplace transform of the modified Bessel function K0?

Note: This is a natural query someone might ask when studying advanced mathematics or signal processing, without knowing about this specific text, and the text contains the mathematical answer to this question.
What is the inverse Laplace transform of the modified Bessel function K0?

Note: This is a natural query someone might ask when studying advanced mathematics or signal processing, without knowing about this specific text, and the text contains the mathematical answer to this question.