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open disks around infinity

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1: 1.9 Calculus of a Complex Variable
It is single-valued on { 0 } , except on the interval ( , 0 ) where it is discontinuous and two-valued. …
Continuity
at ( x , y ) . … A system of open disks around infinity is given by … Suppose n = 0 f n ( t ) converges uniformly in any compact interval in ( a , b ) , and at least one of the following two conditions is satisfied: …
2: 26.15 Permutations: Matrix Notation
If ( j , k ) B , then σ ( j ) k . The number of derangements of n is the number of permutations with forbidden positions B = { ( 1 , 1 ) , ( 2 , 2 ) , , ( n , n ) } . … For ( j , k ) B , B [ j , k ] denotes B after removal of all elements of the form ( j , t ) or ( t , k ) , t = 1 , 2 , , n . B ( j , k ) denotes B with the element ( j , k ) removed. … Let B = { ( j , j ) , ( j , j + 1 ) |  1 j < n } { ( n , n ) , ( n , 1 ) } . …
3: 14.27 Zeros
P ν μ ( x ± i 0 ) (either side of the cut) has exactly one zero in the interval ( , 1 ) if either of the following sets of conditions holds: …For all other values of the parameters P ν μ ( x ± i 0 ) has no zeros in the interval ( , 1 ) . …
4: 1.5 Calculus of Two or More Variables
A function f ( x , y ) is continuous at a point ( a , b ) if … If F ( x , y ) is continuously differentiable, F ( a , b ) = 0 , and F / y 0 at ( a , b ) , then in a neighborhood of ( a , b ) , that is, an open disk centered at a , b , the equation F ( x , y ) = 0 defines a continuously differentiable function y = g ( x ) such that F ( x , g ( x ) ) = 0 , b = g ( a ) , and g ( x ) = F x / F y . … f ( x , y ) has a local minimum (maximum) at ( a , b ) if … Suppose that a , b , c are finite, d is finite or + , and f ( x , y ) , f / x are continuous on the partly-closed rectangle or infinite strip [ a , b ] × [ c , d ) . … Moreover, if a , b , c , d are finite or infinite constants and f ( x , y ) is piecewise continuous on the set ( a , b ) × ( c , d ) , then …
5: 3.5 Quadrature
for some ξ ( a , b ) . … Or if the set x 1 , x 2 , , x n lies in the open interval ( a , b ) , then the quadrature rule is said to be open. … Examples of open rules are the Gauss formulas (§3.5(v)), the midpoint rule, and Fejér’s quadrature rule. … and ξ is some point in ( a , b ) . …
§3.5(x) Cubature Formulas
6: 1.10 Functions of a Complex Variable
We write ( f 1 , D 1 ) , ( f 2 , D 2 ) to signify this continuation. … Suppose f ( z ) is analytic in the annulus r 1 < | z z 0 | < r 2 , 0 r 1 < r 2 , and r ( r 1 , r 2 ) . … If D = ( , 0 ] and z = r e i θ , then one branch is r e i θ / 2 , the other branch is r e i θ / 2 , with π < θ < π in both cases. … (Thus if z 0 is in the interval ( 1 , 1 ) , then the logarithms are real.) … Let F ( x , z ) have a converging power series expansion of the form …
7: 26.6 Other Lattice Path Numbers
D ( m , n ) is the number of paths from ( 0 , 0 ) to ( m , n ) that are composed of directed line segments of the form ( 1 , 0 ) , ( 0 , 1 ) , or ( 1 , 1 ) . … M ( n ) is the number of lattice paths from ( 0 , 0 ) to ( n , n ) that stay on or above the line y = x and are composed of directed line segments of the form ( 2 , 0 ) , ( 0 , 2 ) , or ( 1 , 1 ) . … N ( n , k ) is the number of lattice paths from ( 0 , 0 ) to ( n , n ) that stay on or above the line y = x , are composed of directed line segments of the form ( 1 , 0 ) or ( 0 , 1 ) , and for which there are exactly k occurrences at which a segment of the form ( 0 , 1 ) is followed by a segment of the form ( 1 , 0 ) . … r ( n ) is the number of paths from ( 0 , 0 ) to ( n , n ) that stay on or above the diagonal y = x and are composed of directed line segments of the form ( 1 , 0 ) , ( 0 , 1 ) , or ( 1 , 1 ) . …
26.6.9 n = 0 r ( n ) x n = 1 x 1 6 x + x 2 2 x .
8: 14.16 Zeros
where m , n and δ μ , δ ν ( 0 , 1 ) . … The zeros of 𝖰 ν μ ( x ) in the interval ( 1 , 1 ) interlace those of 𝖯 ν μ ( x ) . … P ν μ ( x ) has exactly one zero in the interval ( 1 , ) if either of the following sets of conditions holds: … For all other values of μ and ν (with ν 1 2 ) P ν μ ( x ) has no zeros in the interval ( 1 , ) . 𝑸 ν μ ( x ) has no zeros in the interval ( 1 , ) when ν > 1 , and at most one zero in the interval ( 1 , ) when ν < 1 .
9: 4.37 Inverse Hyperbolic Functions
In (4.37.2) the integration path may not pass through either of the points ± 1 , and the function ( t 2 1 ) 1 / 2 assumes its principal value when t ( 1 , ) . … It should be noted that the imaginary axis is not a cut; the function defined by (4.37.19) and (4.37.20) is analytic everywhere except on ( , 1 ] . … On the part of the cut from to 1
10: 18.40 Methods of Computation
Let x ( a , b ) . … Here x ( t , N ) is an interpolation of the abscissas x i , N , i = 1 , 2 , , N , that is, x ( i , N ) = x i , N , allowing differentiation by i . …
18.40.9 x ( t , N ) = x 1 , N 1 + a 1 ( t 1 ) 1 + a 2 ( t 2 ) 1 + a N 1 ( t ( N 1 ) ) 1 , t ( 0 , ) ,
The PWCF x ( t , N ) is a minimally oscillatory algebraic interpolation of the abscissas x i , N , i = 1 , 2 , , N . … Further, exponential convergence in N , via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate w ( x ) for these OP systems on x [ 1 , 1 ] and ( , ) respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …