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1: 1.4 Calculus of One Variable
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: 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 . …
3: 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.
  • 4: 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 . …
    5: 4.15 Graphics
    See accompanying text
    Figure 4.15.4: arctan x and arccot x . … arccot x is discontinuous at x = 0 . Magnify
    6: 4.24 Inverse Trigonometric Functions: Further Properties
    7: 4.2 Definitions
    Consequently ln z is two-valued on the cut, and discontinuous across the cut. … This is an analytic function of z on ( - , 0 ] , and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless a . …
    8: 28.12 Definitions and Basic Properties
    As a function of ν with fixed q ( 0 ), λ ν ( q ) is discontinuous at ν = ± 1 , ± 2 , . …
    9: Bibliography D
  • T. M. Dunster (1996b) Asymptotics of the generalized exponential integral, and error bounds in the uniform asymptotic smoothing of its Stokes discontinuities. Proc. Roy. Soc. London Ser. A 452, pp. 1351–1367.
  • 10: 4.37 Inverse Hyperbolic Functions