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1—10 of 101 matching pages
1: Sidebar 5.SB1: Gamma & Digamma Phase Plots
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►Phase changes around the zeros are of opposite sign to those around the poles.
The fluid flow analogy in this case involves a line of vortices of alternating sign of circulation, resulting in a near cancellation of flow far from the real axis.
2: David M. Bressoud
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► 227, in 1980, Factorization and Primality Testing, published by Springer-Verlag in 1989, Second Year Calculus from Celestial Mechanics to Special
Relativity, published by Springer-Verlag in 1992, A Radical
Approach to Real Analysis, published by the Mathematical Association of America in 1994, with a second edition in 2007, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, published by the Mathematical Association of America and Cambridge University Press in 1999, A Course in Computational Number Theory (with S.
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3: 14.27 Zeros
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(a)
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, , , and and have opposite signs.
4: 36.5 Stokes Sets
5: 1.11 Zeros of Polynomials
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►A monic polynomial of even degree with real coefficients has at least two zeros of opposite signs when the constant term is negative.
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Descartes’ Rule of Signs
►The number of positive zeros of a polynomial with real coefficients cannot exceed the number of times the coefficients change sign, and the two numbers have same parity. A similar relation holds for the changes in sign of the coefficients of , and hence for the number of negative zeros of . … ►Both polynomials have one change of sign; hence for each polynomial there is one positive zero, one negative zero, and six complex zeros. …6: 4.16 Elementary Properties
7: 25.10 Zeros
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►Because , vanishes at the zeros of , which can be separated by observing sign changes of .
Because changes sign infinitely often, has infinitely many zeros with real.
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►By comparing with the number of sign changes of we can decide whether has any zeros off the line in this region.
Sign changes of are determined by multiplying (25.9.3) by to obtain the Riemann–Siegel formula:
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8: 32.11 Asymptotic Approximations for Real Variables
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(c)
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If , then changes sign once, from positive to negative, as passes from to .
32.11.10
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32.11.11
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32.11.17
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32.11.21
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9: 14.16 Zeros
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has zeros in the interval , where can take one of the values , , , , subject to being even or odd according as and have opposite signs or the same sign.
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(a)
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, , , and and have opposite signs.