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Integrals and series. Vol. 4: Direct Laplace transforms. (English) Zbl 0786.44003

New York: Gordon and Breach Science Publishers. xx, 619 p. (1992).
This book comprises a very extensive list of direct Laplace transforms, including many published for the first time. The tabulation follows the conventional “original function in the left-, transformation in the right-hand columns”, whilst the required formulae can be found from a detailed contents list at the front of the book.
Most of the functions associated with the theory of operational calculus are represented, including the theta functions, but not the Jacobian elliptic functions, although a reference is given for these latter doubly periodic functions in a comprehensive bibliography at the end of the book.
Of passing interest is the way periodicity can be imposed on representative function segments by replacing the argument with its entier, i.e. for arbitrary \(f(t)\), \(f([t])\) is periodic. This idea figures quite prominently throughout the tables.
Finally, an appendix summarises the properties and applications of Laplace transforms, and there is a particularly edifying bit on the solution of differential and integral equations by the method of Laplace transforms.

MSC:

44A10 Laplace transform
44-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to integral transforms
44A35 Convolution as an integral transform
00A22 Formularies
44A20 Integral transforms of special functions
33-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to special functions
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Digital Library of Mathematical Functions:

§10.22(vi) Compendia ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.43(vi) Compendia ‣ §10.43 Integrals ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions
§10.71(iii) Compendia ‣ §10.71 Integrals ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions
§11.10(x) Integrals and Sums ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
§1.14(viii) Compendia ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods
§11.7(v) Compendia ‣ §11.7 Integrals and Sums ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions
§11.9(iv) References ‣ §11.9 Lommel Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
Nicholson-type Integral ‣ §12.12 Integrals ‣ Properties ‣ Chapter 12 Parabolic Cylinder Functions
Loop Integrals ‣ §13.10(ii) Laplace Transforms ‣ §13.10 Integrals ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions
§13.10(vi) Other Integrals ‣ §13.10 Integrals ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions
§13.23(i) Laplace and Mellin Transforms ‣ §13.23 Integrals ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions
§14.17(v) Laplace Transforms ‣ §14.17 Integrals ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions
Integration by Parts ‣ §1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods
§15.14 Integrals ‣ Properties ‣ Chapter 15 Hypergeometric Function
§16.20 Integrals and Series ‣ Meijer 𝐺-Function ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function
§16.5 Integral Representations and Integrals ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function
§18.17(ix) Compendia ‣ §18.17 Integrals ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials
§19.13(iii) Laplace Transforms ‣ §19.13 Integrals of Elliptic Integrals ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals
§20.10(iii) Compendia ‣ §20.10 Integrals ‣ Properties ‣ Chapter 20 Theta Functions
§24.13(iii) Compendia ‣ §24.13 Integrals ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials
§25.11(ix) Integrals ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
Integral Representation ‣ §25.12(ii) Polylogarithms ‣ §25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
§25.7 Integrals ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
de Branges–Wilson Beta Integral ‣ §5.13 Integrals ‣ Properties ‣ Chapter 5 Gamma Function
§6.14(iii) Compendia ‣ §6.14 Integrals ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals
§7.14(iii) Compendia ‣ §7.14 Integrals ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals
§8.14 Integrals ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions
§9.10(ix) Compendia ‣ §9.10 Integrals ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions