iv CONTENTS
4.4 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5 Partitions into groups 27
5.1 De…nition of partition into groups . . . . . . . . . . . . . . . . . . . 27
5.2 Number of partitions into groups . . . . . . . . . . . . . . . . . . . . 28
5.3 More details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.3.1 Multinomial expansions . . . . . . . . . . . . . . . . . . . . . 29
5.4 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6 Sequences and limits 31
6.1 De…nition of sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.2 Countable and uncountable sets . . . . . . . . . . . . . . . . . . . . . 32
6.3 Limit of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.3.1 The limit of a sequence of real numbers . . . . . . . . . . . . 32
6.3.2 The limit of a sequence in general . . . . . . . . . . . . . . . 34
7 Review of di¤erentiation rules 39
7.1 Derivative of a constant function . . . . . . . . . . . . . . . . . . . . 39
7.2 Derivative of a power function . . . . . . . . . . . . . . . . . . . . . . 39
7.3 Derivative of a logarithmic function . . . . . . . . . . . . . . . . . . . 40
7.4 Derivative of an exponential function . . . . . . . . . . . . . . . . . . 40
7.5 Derivative of a linear combination . . . . . . . . . . . . . . . . . . . 41
7.6 Derivative of a product of functions . . . . . . . . . . . . . . . . . . . 41
7.7 Derivative of a composition of functions . . . . . . . . . . . . . . . . 42
7.8 Derivatives of trigonometric functions . . . . . . . . . . . . . . . . . 43
7.9 Derivative of an inverse function . . . . . . . . . . . . . . . . . . . . 43
8 Review of integration rules 45
8.1 Inde…nite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
8.1.1 Inde…nite integral of a constant function . . . . . . . . . . . . 46
8.1.2 Inde…nite integral of a power func tion . . . . . . . . . . . . . 46
8.1.3 Inde…nite integral of a logarithmic function . . . . . . . . . . 47
8.1.4 Inde…nite integral of an exponential function . . . . . . . . . 47
8.1.5 Inde…nite integral of a linear combination of functions . . . . 47
8.1.6 Inde…nite integrals of trigonometric functions . . . . . . . . . 48
8.2 De…nite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
8.2.1 Fundamental theorem of calculus . . . . . . . . . . . . . . . . 48
8.2.2 De…nite integral of a linear combination of functions . . . . . 49
8.2.3 Change of variable . . . . . . . . . . . . . . . . . . . . . . . . 50
8.2.4 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . 51
8.2.5 Exchanging the b ou nds of integration . . . . . . . . . . . . . 51
8.2.6 Subdividing the integral . . . . . . . . . . . . . . . . . . . . . 51
8.2.7 Leibniz integral rule . . . . . . . . . . . . . . . . . . . . . . . 52
8.3 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
9 Special functions 55
9.1 Gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9.1.1 De…nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9.1.2 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
9.1.3 Relation to the factorial function . . . . . . . . . . . . . . . . 56
9.1.4 Values of the Gamma function . . . . . . . . . . . . . . . . . 57
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