degrees two, three, four
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5 matching pages
1: 1.11 Zeros of Polynomials
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§1.11(iii) Polynomials of Degrees Two, Three, and Four
…2: 18.19 Hahn Class: Definitions
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►The Askey scheme extends the three families of classical OP’s (Jacobi, Laguerre and Hermite) with eight further families of OP’s for which the role of the differentiation operator in the case of the classical OP’s is played by a suitable difference operator.
These eight further families can be grouped in two classes of OP’s:
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1.
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2.
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►The Hahn class consists of four discrete families (Hahn, Krawtchouk, Meixner, and Charlier) and two continuous families (continuous Hahn and Meixner–Pollaczek).
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Hahn class (or linear lattice class). These are OP’s where the role of is played by or or (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.
Wilson class (or quadratic lattice class). These are OP’s ( of degree in , quadratic in ) where the role of the differentiation operator is played by or or . The Wilson class consists of two discrete and two continuous families.
3: Bille C. Carlson
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►Carlson, in 1981, and is survived by his companion, Jody Stadler, two children, Marian Carlson and John Carlson, and four grandchildren.
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►After the war he returned to Harvard and completed Bachelor’s and Master’s degrees in physics and mathematics.
He then went to Oxford as a Rhodes Scholar and completed a doctoral degree in physics.
After four years in the Physics Department at Princeton, he went to the Ames Laboratory and Iowa State University in 1954, where he was a Professor in the Physics and Mathematics Departments.
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►This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions.
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4: 14.15 Uniform Asymptotic Approximations
5: 22.19 Physical Applications
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►The periodicity and symmetry of the pendulum imply that the motion in each four intervals and have the same “quarter periods” .
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►For real and positive, three of the four possible combinations of signs give rise to bounded oscillatory motions.
…The subsequent position as a function of time, , for the three cases is given with results expressed in terms of and the dimensionless parameter .
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►Two types of oscillatory motion are possible.
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►Hyperelliptic functions are solutions of the equation , where is a polynomial of degree higher than 4.
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