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The Laplace Transform

David Vernon Widder
Publisher: 
Dover Publications
Publication Date: 
2010
Number of Pages: 
406
Format: 
Paperback
Price: 
22.95
ISBN: 
9780486477558
Category: 
Monograph
[Reviewed by
Allen Stenger
, on
07/19/2011
]

This monograph, an unaltered reprint of the 1941 edition, is a thorough look at the analytical properties of the Laplace transform. The book develops most of the background needed; it assumes a moderate knowledge of real and complex analysis. The book is aimed at the graduate-student level, but there are no exercises or examples.

The back-cover blurb says the book is “highly theoretical in its emphasis”, which I think is a code-phrase for “no applications”. The book deals primarily with the Laplace transform in isolation, although it does include some applications to other parts of analysis and to number theory. Everything is handled in terms of the (Riemann–)Stieltjes integral, in order to give a unified treatment that covers both integral transforms and generalized Dirichlet series. The book derives all needed properties of these integrals.

Most of the book is aimed at the representation problem (which functions can be expressed as a Laplace transform) and the inversion problem (how to recover a function from its Laplace transform). Leading up to this is a careful study of the Hausdorff moment problem. There is also a long chapter on Tauberian theorems for Laplace transforms, that is not closely related to the rest of the book but is where its applications are. These include the prime number theorem, Littlewood’s strengthening of the original Tauber theorem, and a proof of Wiener’s General Tauberian Theorem.


Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

  • PREFACE
  • CHAPTER I. THE STIELTJES INTEGRAL
    1. Introduction
    2. Stieltjes integrals
    3. Functions of bounded variation
    4. Existence of Stieltjes integrals
    5. Properties of Stieltjes integrals
    6. The Stieltjes integral as a series or a Lebesgue integral
    7. Further properties of Stieltjes integrals
    8. Normalization
    9. Improper Stieltjes integrals
    10. Laws of the mean
    11. Change of variable
    12. Variation of the indefinite integral
    13. Stieltjes integrals as infinite series; second method
    14. Further conditions of integrability
    15. Iterated integrals
    16. The selection principle
    17. Weak compactness
  • CHAPTER II. FUNDAMENTAL FORMULAS
    1. Region of convergence
    2. Abscissa of convergence
    3. Absolute convergence
    4. Uniform convergence
    5. Analytic character of the generating function
    6. Uniqueness of determining function
    7. Complex inversion formula
    8. Integrals of the determining function
    9. Summability of divergent integrals
    10. Inversion when the determining function belongs to L2
    11. Stieltjes resultant
    12. Classical resultant
    13. Order on vertical lines
    14. Generating function analytic at infinity
    15. Periodic determining function
    16. Relation to factorial series
  • CHAPTER III. THE MOMENT PROBLEM
    1. Statement of the problem
    2. Moment sequence
    3. An inversion operator
    4. Completely monotonic sequences
    5. Function of Lp
    6. Bounded functions
    7. Hausdorff summability
    8. Statement of further moment problems
    9. The moment operator
    10. The Hamburger moment problem
    11. Positive definite sequences
    12. Determinant criteria
    13. The Stieltjes moment problem
    14. Moments of functions of bounded variation
    15. A sufficient condition for the solubility of the Stieltjes problem
    16. Indeterminacy of solution
  • CHAPTER IV. ABSOLUTELY AND COMPLETELY MONOTONIC FUNCTIONS
    1. Introduction
    2. Elementary properties of absolutely monotonic functions
    3. Analyticity of absolutely monotonic functions
    4. Bernstein’s second definition
    5. Existence of one-sided derivatives
    6. Higher differences of absolutely monotonic functions
    7. Equivalence of Bernstein’s two definitions
    8. Bernstein polynomials
    9. Definition of Grüss
    10. Equivalence of Bernstein and Grüss definitions
    11. Additional properties of absolutely monotonic functions
    12. Bernstein’s theorem
    13. Alternative proof of Bernstein’s theorem
    14. Interpolation by completely monotonic functions
    15. Absolutely monotonic functions with prescribed derivatives at a point
    16. Hankel determinants whose elements are the derivatives of an absolutely monotonic function
    17. Laguerre polynomials
    18. A linear functional
    19. Bernstein’s theorem
    20. Completely convex functions
  • CHAPTER V. TAUBERIAN THEOREMS
    1. Abelian theorems for the Laplace transform
    2. Abelian theorems for the Stieltjes transform
    3. Tauberian theorems
    4. Karamata’s theorem
    5. Tauberian theorems for the Stieltjes transform
    6. Fourier transforms
    7. Fourier transforms of functions of L
    8. The quotient of Fourier transforms
    9. A special Tauberian theorem
    10. Pitt’s form of Wiener’s theorem
    11. Wiener’s general Tauberian theorem
    12. Tauberian theorem for the Stieltjes integral
    13. One-sided Tauberian condition
    14. Application of Wiener’s theorem to the Laplace transform
    15. Another application
    16. The prime-number theorem
    17. Ikehara’s theorem
  • CHAPTER VI. THE BILATERAL LAPLACE TRANSFORM
    1. Introduction
    2. Region of convergence
    3. Integration by parts
    4. Abscissae of convergence
    5. Inversion formulas
    6. Uniqueness
    7. Summability
    8. Determining function belonging to L2
    9. The Mellin transform
    10. Stieltjes resultant
    11. Stieltjes resultant at infinity
    12. Stieltjes resultant completely defined
    13. Preliminary results
    14. The product of Fourier-Stieltjes transforms
    15. Stieltjes resultant of indefinite integrals
    16. Product of bilateral Laplace integrals
    17. Resultants in a special case
    18. Iterates of the Stieltjes kernel
    19. Representation of functions
    20. Kernels of positive type
    21. Necessary and sufficient conditions for representation
  • CHAPTER VII. INVERSION AND REPRESENTATION PROBLEMS FOR THE LAPLACE TRANSFORM
    1. Introduction
    2. Laplace’s asymptotic evaluation of an integral
    3. Application of the Laplace method
    4. Uniform convergence
    5. Uniform convergence; continuation
    6. The inversion operator for the Laplace-Lebesgue integral
    7. The inversion operator for the Laplace-Stieltjes integral
    8. Laplace method for a new integral
    9. The jump operator
    10. The variation of the determining function
    11. A general representation theorem
    12. Determining function of bounded variation
    13. Modified conditions for determining functions of bounded variation
    14. Determining function non-decreasing
    15. The class Lp, p > 1
    16. Determining function the integral of a bounded function
    17. The class L
    18. The general Laplace-Stieltjes integral
  • CHAPTER VIII. THE STIELTJES TRANSFORM
    1. Introduction
    2. Elementary properties of the transform
    3. Asymptotic properties of Stieltjes transforms
    4. Relation to the Laplace transform
    5. Uniqueness
    6. The Stieltjes transform singular at the origin
    7. Complex inversion formula
    8. A singular integral
    9. The inversion operator for the Stieltjes transform with α(t) an integral
    10. The inversion operator for the Stieltjes transform in the general case
    11. The jump operator
    12. The variation of α(t)
    13. A general representation theorem
    14. Order conditions
    15. General representation theorems
    16. The function α(t) of bounded variation
    17. The function α(t) non-decreasing and bounded
    18. The function α(t) non-decreasing and unbounded
    19. The class Lp, p > 1
    20. The function φ(t) bounded
    21. The class L
    22. The function α(t) of bounded variation in every finite interval
    23. Operational considerations
    24. The iterated Stieltjes transform
    25. Application to the Laplace transform
    26. Solution of an integral equation
    27. A related integral equation

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