## Introduction to the Theory of Analytic Functions (Google eBook) |

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### Common terms and phrases

absolute value absolutely convergent algebraic function amplitude angle assigned bilinear transformation branch-point called Cauchy's centre circle of convergence circuit coefficients complex numbers consider constant continuous convergent series corresponding curve defined definition denote derivate domain elliptic function equal equation essential singular essential singular point example expression finite number formula fractional fundamental region fxdx geometric given Hence imaginary infinite decimal infinity interval inverse irrational number jr-plane Laurent series logarithm lower limit monogenic function namely negative non-essential singular one-valued analytic function one-valued function pairs parallelogram path plane positive number power series prove radius of convergence rational number rational objects real numbers real variable rectangle Riemann surface sequence singular points straight line stroke suppose Taylor's theorem tends th root theory of functions tlie transcendental integral function uniformly convergent upper limit Weierstrass's zero

### Popular passages

Page 315 - An Essay on the application of Mathematical Analysis to the Theories of Electricity and Magnetism...

Page 87 - If in each row of a determinant the absolute value of the element on the principal diagonal is greater than the sum of the absolute values of the remaining elements in that row, the value of the determinant is different from zero. PROOF Let | A \ be an...

Page vii - ... a foundation of algebraic truths. It is therefore not correct to turn around and, expressing myself briefly, use "transcendental...

Page 325 - Also we see that, by making n sufficiently large, we can make the fraction — - as small as we please. Thus by taking a sufficient number of terms the sum can be made to differ by as little as we please from 2. In the next article a more general case is discussed.

Page 67 - P and a neighbouring point on the curve can be made to differ from it by as little as we please...

Page 112 - Sc is also absolutely convergent, and its sum is the product of the sums of the two former series.

Page 14 - ... levers: the first has the fulcrum between the power and weight; in the second the weight acts between the fulcrum and the power; and in the third the power acts between the fulcrum and the weight. PROP. To find the conditions of equilibrium of two forces acting in the same plane on a lever. 93. Let the plane of the...

Page 4 - ... and not before the other. It is very important to notice that we have now a closed number-system. When we seek to separate the irrational objects as lying left or right of an object, either the object is rational, or if not it separates rational objects and is irrational ; in any case...

Page 123 - The idea that series of powers are as serviceable for algebra as for arithmetic was first worked out by Newton*, and in the theory of functions of a complex variable, as it now stands, the theory of such series is the solid foundation for the whole structure.

Page 3 - ... comes first. It is to be noticed that as we approach any of the natural objects there is no last fractional mark ; that is, whatever object we take there are always others between it and the natural object.