The History of Mathematics: A Brief Course

502 17. REAL AND COMPLEX ANALYSIS if r and è are regarded as polar coordinates, while e~ay cos(ax) and e~ay sin(ax) are harmonic ...

REAL ANALYSIS 503 2.3. Uniform convergence and continuity. Cauchy was not aware at first of any need to make the distinction b ...

504 17. REAL AND COMPLEX ANALYSIS justified his more general integral in the following words, from the preface to his 1904 monog ...

REAL ANALYSIS 505 geometric figures, it was necessary to investigate differentiation in more detail as well. The secret of tha ...

506 17. REAL AND COMPLEX ANALYSIS operation still led to the invocation of some strong principles of inference in the context of ...

QUESTIONS AND PROBLEMS 507 A set is of first category if it is the union of a sequence of sets Ak such that every interval (a,b) ...

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Part 7. Mathematical Inferences The History of Mathematics: A Brief Course, Second Edition by Roger Cooke Copyright © 200 5 John ...

At various points in this survey of mathematics we have found mathematicians de- bating the meaning of what they were doing and ...

Chapter 18. Probability and Statistics The need to make decisions on the basis of incomplete data is very widespread in human li ...

512 18. PROBABILITY AND STATISTICS of the Sun, we use a deterministic model (geometric astronomy, in this case) to study and pre ...

PROBABILITY 513 1 2 3 4 5 6 7 8 9 10 11 12 108 111 115 120 126 133 33 36 37 36 33 26 Readers who enjoy playing with numbers ma ...

514 18. PROBABILITY AND STATISTICS the game is terminated after three unsuccessful tries. Pascal wrote to Fermat that the player ...

PROBABILITY 515 1.4. Leibniz. Although Leibniz wrote a full treatise on combinatorics, which pro- vides the mathematical appar ...

516 18. PROBABILITY AND STATISTICS numbers, defined by the formula fe=l Nowadays we define these numbers as Bo = 1, B\ = — |, an ...

PROBABILITY 517 1.6. De Moivre. In 1711, even before the appearance of Jakob Bernoulli's trea- tise, another groundbreaking bo ...

518 18. PROBABILITY AND STATISTICS 25- 20- 15- 10- 5- I 12 3 4 5 6 7 8 9 1011 12131415161718192021222324252627282930313233343536 ...

PROBABILITY 519 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 FIGURE 2. Histogram of the frequencies of the frequencies in ...

520 18. PROBABILITY AND STATISTICS and as a result, the total utility of personal wealth is a logarithmic function of total weal ...

PROBABILITY 521 Except for using the letter c where we now use e to denote the base of natural loga- rithms, he had what we no ...