## A Course of Modern Analysis: An Introduction to the General Theory of Infinite Series and of Analytic Functions, with an Account of the Principal Transcendental Functions |

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### Common terms and phrases

absolutely convergent addition-theorem analytic function ascending powers asymptotic expansion Bessel functions Bessel's equation Cambridge Mathematical Tripos circle coefficients complex number Consider contour convergent series corresponding curve defined definite integral denote divergent doubly-periodic function elliptic function enclosing the point equal expression finite follows formula Fourier series function f(z function sn Gamma-function given Hence hypergeometric function hypergeometric series infinite product infinite series integrand interior Jn(z Laplace's equation last article Legendre functions limit Liouville's theorem moduli multiple negative integer non-uniformly convergent number of terms obtain origin particular solutions period-parallelogram plane poles polynomials positive integer positive quantity power-series Prove radius real axis real values region regular function represents residue result roots satisfied series converges Shew shewn singularities suppose taken round Taylor's Theorem tends to infinity tends to zero uniformly convergent unity variable write z-plane

### Popular passages

Page 90 - The former denotes a family of straight lines whose distance from the origin is equal to a, the latter a circle whose centre is at the origin, and whose radius is equal to a. And here, as was noted generally by Lagrange, the singular solution seems to be, in relation to geometry, the more important of the two. 3. A more general class of problems is that in which it is required to determine the curves in which some one of the foregoing elements, Art. 1, is equal...

Page 313 - This is a linear differential equation of the second order with constant coefficients ; its solution, found in the usual way, is where A and B are arbitrary constants.

Page 40 - Those quantities which retain the same value are called constant; those whose values are varying are called variable. When variable quantities are so connected that the value of one of them is determined by the value ascribed to the others, that variable quantity is said to be a function of the others.

Page 267 - Untersuchung des Theils der planetarischen Storungen, welcher aus der Bewegung der Sonne entsteht.

Page 4 - A complex number is an expression of the form a + bi, where a and b are real numbers and i =V~ 1 -In the complex number, a is called the real part and bi is the imaginary part.

Page 315 - Laplace's equation, and equating to zero the coefficients of the various powers of x, y, and...

Page 6 - I by 6, then r and 6 are clearly the radius vector and vectorial angle of the point P, referred to the origin 0 and axis Ox. The representation of complex quantities thus afforded is often called the Argand diagram*.

Page 110 - Consider now a function /(*), whose only singularities in the finite part of the plane are...

Page 123 - Although the persymmetric determinant where sr is the sum of the rth powers of the roots of the equation x...

Page 182 - ... by the stipulation that the variable is not to cross the real axis at any point on the positive side of the origin.