on intervals
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1—10 of 229 matching pages
1: 7.23 Tables
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Abramowitz and Stegun (1964, Chapter 7) includes , , , 10D; , , 8S; , , 7D; , , , 6S; , , 10D; , , 9D; , , , 7D; , , , , 15D.
Finn and Mugglestone (1965) includes the Voigt function , , , 6S.
Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of , , , 7D and 8D, respectively; the real and imaginary parts of , , , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.
2: 14.27 Zeros
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(either side of the cut) has exactly one zero in the interval
if either of the following sets of conditions holds:
…For all other values of the parameters has no zeros in the interval
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3: 26.15 Permutations: Matrix Notation
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►If , then .
The number of derangements of is the number of permutations with forbidden positions .
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►For , denotes after removal of all elements of the form or , .
denotes with the element removed.
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►Let .
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4: 14.16 Zeros
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§14.16(ii) Interval
… ►The zeros of in the interval interlace those of . … ►§14.16(iii) Interval
► has exactly one zero in the interval if either of the following sets of conditions holds: … ► has no zeros in the interval when , and at most one zero in the interval when .5: 22.17 Moduli Outside the Interval [0,1]
§22.17 Moduli Outside the Interval [0,1]
… ►Jacobian elliptic functions with real moduli in the intervals and , or with purely imaginary moduli are related to functions with moduli in the interval by the following formulas. … ►For proofs of these results and further information see Walker (2003).6: 1.4 Calculus of One Variable
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►Suppose is defined on .
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►A function is convex on if
…for any , and .
…A similar definition applies to closed intervals
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7: 26.6 Other Lattice Path Numbers
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is the number of paths from to that are composed of directed line segments of the form , , or .
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is the number of lattice paths from to that stay on or above the line and are composed of directed line segments of the form , , or .
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is the number of lattice paths from to that stay on or above the line , are composed of directed line segments of the form or , and for which there are exactly occurrences at which a segment of the form is followed by a segment of the form .
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is the number of paths from to that stay on or above the diagonal and are composed of directed line segments of the form , , or .
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8: 4.37 Inverse Hyperbolic Functions
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►In (4.37.2) the integration path may not pass through either of the points , and the function assumes its principal value when .
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4.37.19
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►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 .
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4.37.22
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4.37.24
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9: 1.5 Calculus of Two or More Variables
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►A function is continuous at a point
if
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has a local minimum (maximum) at if
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►for all and all .
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►Let be defined on a closed rectangle .
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►Moreover, if are finite or infinite constants and is piecewise continuous on the set , then
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