## Problems and Theorems in Analysis IFrom the reviews: "The work is one of the real classics of this century; it has had much influence on teaching, on research in several branches of hard analysis, particularly complex function theory, and it has been an essential indispensable source book for those seriously interested in mathematical problems. These volumes contain many extraordinary problems and sequences of problems, mostly from some time past, well worth attention today and tomorrow. Written in the early twenties by two young mathematicians of outstanding talent, taste, breadth, perception, perseverence, and pedagogical skill, this work broke new ground in the teaching of mathematics and how to do mathematical research. (Bulletin of the American Mathematical Society) |

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### Contents

Infinite Series and Infinite Sequences | 1 |

Linear Transformations of Series A Theorem of Cesŕro | 15 |

The Structure of Real Sequences and Series | 23 |

Miscellaneous Problems | 32 |

Integration | 46 |

Inequalities | 62 |

Some Properties of Real Functions | 75 |

Various Types of Equidistribution | 85 |

Functions of Large Numbers | 95 |

Functions of One Complex Variable General Part | 103 |

Mappings and Vector Fields | 113 |

Some Geometrical Aspects of Complex Variables | 125 |

Cauchys Theorem The Argument Principle | 133 |

Sequences of Analytic Functions | 145 |

The Maximum Principle | 157 |

### Common terms and phrases

absolute value absolutely convergent According analytic function Annls Math arbitrary Assume boundary points coefficients complex numbers consider const constant continuous function convex convex function convex polygon curve defined denote differential diverges domain ellipses entire function equal equation equidistributed exists finite formula function f(x G. H. Hardy geometric half-plane Hence identically implies Improper Integrals increase to infinity inequality infinite inner point integer interval limit linear maps maximum monotone increasing non-negative notation obtain Phys plane poles Polya polynomial positive integer positive numbers power series problem proof properly integrable proposition Prove radius radius of convergence real axis region regular resp right hand side schlicht sequence smaller solution subinterval subscript subseries sufficiently Suppose Szego theorem uniformly unit circle unit disk vanish variable vector field winding number z-plane zeros