1.1 Conic sections
1.2 Areas on a sphere
1.3 Lines in a triangle
1.4 Angle multisection
1.5 Prismoidal formula
1.6 Inversion
1.7 Area and volume equivalence
1.8 Isoperimetric problems for convex hulls
1.9 Reflections and images
1.10 Roulettes and pedals
1.11 Integral geometry
1.12 Convexity
1.13 Curves in n dimensions

2 Iteration 51

2.1 Preliminaries
2.2 Explicit iteration and conjugacy
2.3 Duplication and triplication
2.4 Extension to noninteger iteration
2.5 Newton-Raphson method and iteration
2.6 Approximate iteration
2.7 Multiple recursions
2.8 An application to probability
2.9 Iteration of a function of several variables
2.10 Iteration and orders of magnitude
2.11 Iteration and reliable circuits

3 Series and Products 81

3.1 Telescoping cancellation
3.2 Euler's identity and Veita's formula
3.3 Euler's products and Cantor's theorem
3.4 Telescoping coincidence
3.5 Summation of certain series
3.6 Products and factorization
3.7 Infinite products and Gamma-function
3.8 The McLaurin series for tan x and sec x
3.9 Euler acceleration and the Taylor series
3.10 Generalization of Taylor's series
3.11 Blissard-Licas calculus and the Bernoulli numbers
3.12 Poisson summation formula
3.13 Darboux and Euler-McLaurin formulas
3.14 Lambert series

4 Miscellaneous Elementary Topics 131

4.1 Choosing a suitable additional unknown
4.2 Multiplications, additions, and complexity
4.3 Polygonal paths and minimization
4.4 Some types of symmetry
4.5 Extremal problems and extremal points
4.6 Circles on a torus
4.7 Composite estimates
4.8 Sequential localization and maxima of unimodal surfaces
4.9 Working backward from the end and working from both ends
4.10 Computing pi
4.11 The generalized firing squad synchronization and the labyrinth problems

5 Integrals 175

5.1 Integrals evaluable by Fubini's theorem
5.2 Differentiation with respect to a parameter
5.3 Integrals evaluable from first principles
5.4 Dilogarithms
5.5 Crofton's theorem and configuration averages
5.6 Forbidden configurations and discontinuous factors
5.7 Miscellaneous
5.8 Integration and Mensuration
5.9 Reduction of multiple integrals to single ones
5.10 Integration as a source of new functions

6 Miscellaneous Intermediate Topics 213

6.1 Principle of specialization and generating function
6.2 Irrationality proofs
6.3 Partitions and expanding infinite products in series
6.4 Principle of infinite crowding
6.5 Applications of certain special functions
6.6 Continuation principle
6.7 Asymptotic analysis
6.8 Coincidences, forbidden configurations, and hypergraphs

Bibliography 261

Index 265