##
**Table of definite and infinite integrals.**
*(English)*
Zbl 0556.44003

Physical Sciences Data, 13. Amsterdam - Oxford - New York: Elsevier Scientific Publishing Company. X, 457 p. $ 106.50; Dfl. 250.00 (1983).

This table is a collection of integrals of elementary and special functions and it can be viewed as a useful addition to the well-known existing tables. The integrals presented here were compiled over many years from numerous scientific and technical journals and books or were obtained by the author himself. The result is a remarkable collection. There are slightly altered versions of well-known integrals. An example is on p.20, no. 24, which is obtained just by using the property \(K_{\nu}(a) = K_{-\nu}(a)\) for the Bessel function. (A further change \(\nu \to \nu +1\) yields an error in no.25). An eye-catcher is no. 1 on p.48, which says that \(\int^{\infty}_{0}\tan (x+ia) dx=i\pi\) (Re \(a>0)\), but the integral makes no sense to me.

Another one is no.22 on p.104, saying that \(\int^{\infty}_{0}x^{n-} \ln x \sec h x dx=0,\) with no restrictions on n (but I suppose \(n=0,1,2,...)\). Is this correct? Of course it is not fair to consider just examples where the book is wrong. A random choice of, say, 100 items and check these ones would be a better treatment. However, although some of the results are easily verified, the bulk of the book is not, and verification of these 100 items would be very time-consuming. In the same manner, the book may be very useful for mathematicians, engineers and scientists, since it may save them lots of hours when they find the integrals they use in closed form. In any book of special functions errors will occur. I agree with Askey when he said that a formula in any such book may be a good start and always should be verified.

The presentation is clear and the table is well organized.

Another one is no.22 on p.104, saying that \(\int^{\infty}_{0}x^{n-} \ln x \sec h x dx=0,\) with no restrictions on n (but I suppose \(n=0,1,2,...)\). Is this correct? Of course it is not fair to consider just examples where the book is wrong. A random choice of, say, 100 items and check these ones would be a better treatment. However, although some of the results are easily verified, the bulk of the book is not, and verification of these 100 items would be very time-consuming. In the same manner, the book may be very useful for mathematicians, engineers and scientists, since it may save them lots of hours when they find the integrals they use in closed form. In any book of special functions errors will occur. I agree with Askey when he said that a formula in any such book may be a good start and always should be verified.

The presentation is clear and the table is well organized.

Reviewer: N.M.Temme

### MSC:

44A20 | Integral transforms of special functions |

00A22 | Formularies |

44-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to integral transforms |

33-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to special functions |

33-XX | Special functions |

26A09 | Elementary functions |

26A42 | Integrals of Riemann, Stieltjes and Lebesgue type |

65A05 | Tables in numerical analysis |

### 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

§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

§12.5(iv) Compendia ‣ §12.5 Integral Representations ‣ Properties ‣ Chapter 12 Parabolic Cylinder Functions

§13.10(vi) Other Integrals ‣ §13.10 Integrals ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions

§13.23(v) Other Integrals ‣ §13.23 Integrals ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric 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

§4.10(iii) Compendia ‣ §4.10 Integrals ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions

§4.26(v) Compendia ‣ §4.26 Integrals ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions

§4.40(v) Compendia ‣ §4.40 Integrals ‣ Hyperbolic Functions ‣ Chapter 4 Elementary 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

§8.19(x) Integrals ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions