具体数学是与离散数学正好相对应的数学学科的分支。 具体数学和离散数学一样也是计算机科学的不可分割的一部分，应用于程序设计和算法式分析。我们所说的具体数学实际上是指《Concrete Mathematics: A Foundation for Computer Science, 2/E》一书，以下为此书内容介绍。
内容简介This book introduces the mathematics that supports advanced computer Programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills--the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle Patterns in data. It is an indispensable text and reference not only for computer scientists--the authors themselves rely heavily on it! but for serious users Of mathematics in virtually every discipline.
Concrete mathematics is a blending of continuous and disCRETE mathematics: "More concretely," the authors explain, "it is the controlled manipulation of mathematical formulas,using a collection of techniques for solving problems." The subject mater is primarily an expansion of the Mathematical Preliminaries section in Knuth's c1assic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study.
——来自《具体数学（第二版）》英文版
《具体数学》第二版暂无中文，但第一版中文版在网上可以找到pdg的，清晰完全可以接受。本书可以作为研究高德纳(Donal E.Knuth)的另一部著作《计算机程序设计艺术》的前期数学准备，亦是第一章的扩充。
本书讨论范围有 递归、和、整函数、数论、二项系数、特殊数、母函数、离散概率、渐进，这些都是程序设计中所要涉及的基础数学知识，掌握好对今后的程序设计有莫大的帮助。
作者Ronald L. Grahamis chief Scientist at At&T Lab research.
Donal E.Knuthis Professor Emeritus of the Art of Computer programming at Stanford University.
Oren Patashnikis a member of the researth staff at the centor for Communitcation Research, La Jolla.
Donald E. Knuth is known throughout the world for his pioneering work on algorithms and programming techniques, for his invention of the Tex and Metafont systems for computer typesetting, and for his prolific and influential writing. Professor Emeritus of The Art of Computer Programming at Stanford University, he currently devotes full time to the completion of these fascicles and the seven volumes to which they belong.
目录1 Recurrent Problems 1
1.1 The Tower of Hanoi 1
1.2 Lines in the Plane 4
1.3 The Josephus Problem 8
Exercises 17
2 Sums 21
2.1 Notation 21
2.2 Sums and Recurrences 25
2.3 Manipulation of Sums 30
2.4 Multiple Sums 34
2.5 General Methods 41
2.6 Finite and Infinite Calculus 47
2.7 Infinite Sums 56
Exercises 62
3 Integer Functions 67
3.1 Floors and Ceilings 67
3.2 Floor/Ceiling Applications 70
3.3 Floor/Ceiling Recurrences 78
3.4 !(R)mod! The Binary Operation 81
3.5 Floor/Ceiling Sums 86
Exercises 95
4 Number Theory 102
4.1 Divisibility 102
4.2 Primes 105
4.3 Prime Examples 107
4.4 Factorial Factors 111
4.5 Relative Primality 115
4.6 !(R)mod!ˉ: The Congruence Relati 123
4.7 Independent Residues 126
4.8 Additional Applications 129
4.9 Phi and Mu 133
Exercises 144
5 Binomial Coefficients 153
5.1 Basic Identities 153
5.2 Basic Practice 172
5.3 Tricks of the Trade 186
5.4 Generating Functions 196
5.5 Hypergeometric Functions 204
5.6 Hypergeometric Transformations 216
5.7 Partial Hypergeometric Sums 223
Exercises 230
6 Special Numbers
6.1 Stirling Numbers 243
6.2 Eulerian Numbers 253
6.3 Harmonic Numbers 258
6.4 Harmonic Summation 265
6.5 Bernoulli Numbers 269
6.6 Fibonacci Numbers 276
6.7 Continuants 287
Exercises 295
7 Generating Functions
7.1 Domino Theory and Change 306
7.2 Basic Maneuvers 317
7.3 Solving Recurrences 323
7.4 Special Generating Functions 336
7.5 Convolutions 339
7.6 Exponential Generating Functions 350
7.7 Dirichlet Generating Functions 356
Exercises 357
8 Discrete Probability
8.1 Definitions 367
8.2 Mean and Variance 373
8.3 Probability Generating Functions 380
8.4 Flipping Coins 387
8.5 Hashing 397
Exercises 413
9 Asymptotics
9.1 A Hierarchy 426
9.2 0 Notation 429
9.3 0 Manipulation 436
9.4 Two Asymptotic Tricks 449
9.5 Euler!ˉs Summation Formul 455
9.6 Final Summations 462
Exercises 475
A Answers to Exercises
B Bibliography
C Credits for Exercises
Index
List of Tables
