scholarly journals The Product of Two Legendre Polynomials

1953 ◽  
Vol 1 (3) ◽  
pp. 121-125 ◽  
Author(s):  
John Dougall
Keyword(s):  
Linear Function ◽  
Spherical Harmonics ◽  
Legendre Polynomials ◽  
Image Position ◽  
Two Sides

1. It is known that any polynomial in μ. can be expanded as a linear function of Legendre polynomials [1]. In particular, we haveThe earlier coefficients, say A0, A2, A4 may easily be found by equating the coefficients of μp+q, μp+q-2, μp+q-4 on the two sides of (1). The general coefficient A2k might then be surmised, and the value verified by induction. This may have been the method followed by Ferrers, who stated the result as an exercise in his Spherical Harmonics (1877). A proof was published by J. C. Adams [2]. The proof now to be given follows different lines from his.

scholarly journals Derivatives of addition theorems for Legendre functions

1995 ◽  
Vol 37 (2) ◽  
pp. 212-234 ◽  
Author(s):  
D.E. Winch ◽  
P.H. Roberts
Keyword(s):  
Spherical Harmonics ◽  
Chebyshev Polynomials ◽  
Legendre Polynomials ◽  
Addition Theorem ◽  
Legendre Functions ◽  
Addition Theorems ◽  
Associated Legendre Functions ◽  
Tensor Spherical Harmonics ◽  
Derivatives Of

AbstractDifferentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.

scholarly journals The Chaetotaxy of the Pupa of Stegomyia fasciata

1920 ◽  
Vol 10 (2) ◽  
pp. 161-169 ◽  
Author(s):  
J. W. S. Macfie
Keyword(s):  
The Body ◽  
Posterior Angle ◽  
Image Position ◽  
The Right ◽  
Two Sides

The pupa is bilaterally symmetrical, that is, setae occur in similar situations on each side of the body, so that it will suffice to describe the arrangement on one side only. The setae on the two sides of the same pupa, however, often vary as regards their sub-divisions, and similar variations occur between different individuals; as an example, in Table I are shown some of the variations that were found in ten pupae taken at random. An examination of a larger number would have revealed a wider range. As a rule, a seta which is sometimes single, sometimes divided, is longer when single. For example, in one pupa the seta at the posterior angle ofthe seventh segment was single on the right side, double on the left; the former measuring 266μ, and the latter only 159μ in length. This fact is not specifically mentioned in the descriptions which follow, but should be understood.

Summability of the Heine and Neumann Series of Legendre Polynomials

1966 ◽  
Vol 18 ◽  
pp. 1261-1263 ◽  
Author(s):  
Amnon Jakimovski
Keyword(s):  
Holomorphic Function ◽  
Jordan Curve ◽  
Legendre Polynomials ◽  
Closed Interval ◽  
Neumann Series ◽  
Image Position

With a holomorphic function f(z) defined in a domain H which includes the closed interval [—1, 1] we associate the Neumann series1where Pn(z), Qn(t) are, respectively, the nth Legendre polynomials of the first and second kind and γ is a closed and rectifiable Jordan curve which includes [— 1, 1] in its interior and is included, together with its interior, in H.

The Two-Sided Factorization of Ordinary Differential Operators

1980 ◽  
Vol 32 (5) ◽  
pp. 1045-1057 ◽  
Author(s):  
Patrick J. Browne ◽  
Rodney Nillsen
Keyword(s):  
Differential Operator ◽  
Linear Function ◽  
Bilinear Form ◽  
Differential Operators ◽  
Real Numbers ◽  
The Real ◽  
Differentiable Functions ◽  
Ordinary Differential Operators ◽  
Image Position

Throughout this paper we shall use I to denote a given interval, not necessarily bounded, of real numbers and Cn to denote the real valued n times continuously differentiable functions on I and C0 will be abbreviated to C. By a differential operator of order n we shall mean a linear function L:Cn → C of the form1.1where pn(x) ≠ 0 for x ∊ I and pi ∊ Cj 0 ≦ j ≦ n. The function pn is called the leading coefficient of L.It is well known (see, for example, [2, pp. 73-74]) thai a differential operator L of order n uniquely determines both a differential operator L* of order n (the adjoint of L) and a bilinear form [·,·]L (the Lagrange bracket) so that if D denotes differentiation, we have for u, v ∊ Cn,1.2

A Relation Between Ultraspherical and Jacobi Polynomial Sets

1953 ◽  
Vol 5 ◽  
pp. 301-305 ◽  
Author(s):  
Fred Brafman
Keyword(s):  
Jacobi Polynomial ◽  
Legendre Polynomials ◽  
Jacobi Polynomials ◽  
Ultraspherical Polynomials ◽  
Image Position ◽  
Special Case

The Jacobi polynomials may be defined bywhere (a)n = a (a + 1) … (a + n — 1). Putting β = α gives the ultraspherical polynomials which have as a special case the Legendre polynomials .

scholarly journals The Asymptotic Expansions of the Spherical Harmonics

1922 ◽  
Vol 41 ◽  
pp. 82-93
Author(s):  
T. M. MacRobert
Keyword(s):  
Real Axis ◽  
Spherical Harmonics ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
The Real ◽  
Image Position

Associated Legendre Functions as Integrals involving Bessel Functions. Let,where C denotes a contour which begins at −∞ on the real axis, passes positively round the origin, and returns to −∞, amp λ=−π initially, and R(z)>0, z being finite and ≠1. [If R(z)>0 and z is finite, then R(z±)>0.] Then if I−m (λ) be expanded in ascending powers of λ, and if the resulting expression be integrated term by term, it is found that

The triode oscillation generator and amplifier: limitations on sinoidal performance

1930 ◽  
Vol 26 (4) ◽  
pp. 507-527
Author(s):  
L. B. Turner ◽  
L. A. Meacham
Keyword(s):  
Linear Function ◽  
Fundamental Frequency ◽  
Alternating Current ◽  
Anode Potential ◽  
Anode Current ◽  
Simple Analysis ◽  
Oscillation Generator ◽  
Image Position ◽  
Characteristic Relationship ◽  
Grid Potential

The characteristic relationship subsisting between anode current ia, anode potential ea, and grid potential eg in a well-evacuated thermionic triode, viz.leads to very simple analysis of the circuit conditions when μ, is a constant and φ is a linear function. In all the usual applications of triodes, μ is very nearly a constant; in some applications, e.g. in good acoustic amplifiers, over the working range φ is very nearly linear; in others, e.g. in rectifiers, non-linearity is essential to the performance. In oscillators, and in power amplifiers used with them, both conditions are met. Where a sinoidal oscillation is desired, i.e. an alternating current as devoid of harmonics of the fundamental frequency as can be contrived, the working range must be sensibly linear. A feature of such a régime is that the major portion of the power supplied to the triode must be dissipated in heating the anode. Where higher efficiency is required, the cycle must be made to extend beyond the linear range, and harmonics are necessarily introduced.

An extension of the Ahlfors distortion theorem

1954 ◽  
Vol 50 (2) ◽  
pp. 261-265
Author(s):  
F. Huckemann
Keyword(s):  
Conformal Mapping ◽  
Real Axis ◽  
Upper Bounds ◽  
Distortion Theorem ◽  
Lower And Upper Bounds ◽  
The Real ◽  
Strip Domain ◽  
Parallel Strip ◽  
Image Position ◽  
Two Sides

1. The conformal mapping of a strip domain in the z-plane on to a parallel strip— parallel, say, to the real axis of the ζ ( = ξ + iμ)-plane—brings about a certain distortion. More precisely: consider a cross-cut on the line ℜz = c joining the two sides of the frontier of the strip domain (in these introductory remarks we suppose for simplicity that there is only one such cross-cut on that line), and denote by ξ1(c) and ξ2(c) the lower and upper bounds of ξ on the image in the ζ-plane. The theorem of Ahlfors (1), now classical, states thatprovided thatwhere a is the width of the parallel strip and θ(c) the length of the cross-cut.

The extremal values of Legendre polynomials and of certain related functions

1950 ◽  
Vol 46 (4) ◽  
pp. 549-554 ◽  
Author(s):  
R. Cooper
Keyword(s):  
Legendre Polynomials ◽  
Extreme Value ◽  
Positive Zero ◽  
Right Hand ◽  
Extremal Values ◽  
Image Position ◽  
The Right

1. The tabulated values of the Legendre polynomials suggest that the right-hand minimum of Pn(x) changes monotonically as n increases. Let xr, n be the value of x which gives the rth extreme value to the left of 1 of Pn(x). Then we can show thatwhere jr is the rth pösitive zero of J1(z), and that after some term the sequence Pn(xr, n) is monotonic with the moduli of the terms decreasing. We cannot, however, show that the sequence is monotonic from the place at which its terms become significant.

Sign in / Sign up

Export Citation Format

Share Document