## 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

a₁ absolutely convergent analytic function ascending powers asymptotic expansion b₁ b₂ Bessel's equation Cambridge Mathematical Tripos circle coefficients constant contour convergent series defined definite integral denote dz dz elliptic function enclosing the point Example expression finite formula Fourier series function f(z Gamma-function Hence hypergeometric differential equation hypergeometric function hypergeometric series infinite product integer integrand Legendre differential equation Legendre functions Limit multiple obtain P₁ P₂ Pn+1 poles polynomials positive integer power-series Prove quantity radius real axis real values represents residue result satisfied series converges Shew shewn sin² singularities sn² suppose taken round tends to infinity tends to zero theorem U₂ Un+1 unity Up₂ variable w₁ w₂ write z-plane π π

### Popular passages

Page 82 - 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 305 - 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 32 - 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 259 - 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 307 - 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 102 - Consider now a function /(*), whose only singularities in the finite part of the plane are...

Page 115 - Although the persymmetric determinant where sr is the sum of the rth powers of the roots of the equation x...

Page 174 - ... by the stipulation that the variable is not to cross the real axis at any point on the positive side of the origin.