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removable discontinuity

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1: 1.4 Calculus of One Variable
A removable singularity of f ( x ) at x = c occurs when f ( c + ) = f ( c - ) but f ( c ) is undefined. … A simple discontinuity of f ( x ) at x = c occurs when f ( c + ) and f ( c - ) exist, but f ( c + ) f ( c - ) . If f ( x ) is continuous on an interval I save for a finite number of simple discontinuities, then f ( x ) is piecewise (or sectionally) continuous on I . For an example, see Figure 1.4.1
2: Viewing DLMF Interactive 3D Graphics
Any installed VRML or X3D browser should be removed before installing a new one. …
3: 5.10 Continued Fractions
4: 10.25 Definitions
In particular, the principal branch of I ν ( z ) is defined in a similar way: it corresponds to the principal value of ( 1 2 z ) ν , is analytic in ( - , 0 ] , and two-valued and discontinuous on the cut ph z = ± π . … The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in ( - , 0 ] , and two-valued and discontinuous on the cut ph z = ± π . …
5: 1.10 Functions of a Complex Variable
This singularity is removable if a n = 0 for all n < 0 , and in this case the Laurent series becomes the Taylor series. …Lastly, if a n 0 for infinitely many negative n , then z 0 is an isolated essential singularity. … An isolated singularity z 0 is always removable when lim z z 0 f ( z ) exists, for example ( sin z ) / z at z = 0 . … A cut domain is one from which the points on finitely many nonintersecting simple contours (§1.9(iii)) have been removed. … Branches of F ( z ) can be defined, for example, in the cut plane D obtained from by removing the real axis from 1 to and from - 1 to - ; see Figure 1.10.1. …
6: 25.14 Lerch’s Transcendent
25.14.1 Φ ( z , s , a ) n = 0 z n ( a + n ) s , | z | < 1 ; s > 1 , | z | = 1 .
7: 26.15 Permutations: Matrix Notation
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. …
8: 23.18 Modular Transformations
23.18.7 s ( d , c ) = r = 1 c - 1 r c ( d r c - d r c - 1 2 ) , c > 0 .
9: Bibliography W
  • R. Wong and Y.-Q. Zhao (1999a) Smoothing of Stokes’s discontinuity for the generalized Bessel function. II. Proc. Roy. Soc. London Ser. A 455, pp. 3065–3084.
  • R. Wong and Y.-Q. Zhao (1999b) Smoothing of Stokes’s discontinuity for the generalized Bessel function. Proc. Roy. Soc. London Ser. A 455, pp. 1381–1400.
  • 10: 10.2 Definitions
    Except in the case of J ± n ( z ) , the principal branches of J ν ( z ) and Y ν ( z ) are two-valued and discontinuous on the cut ph z = ± π ; compare §4.2(i). … The principal branches of H ν ( 1 ) ( z ) and H ν ( 2 ) ( z ) are two-valued and discontinuous on the cut ph z = ± π . …