THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS OF A REAL VARIABLE

2012 ◽  
pp. 398-446
Author(s):  
G. H. Hardy ◽  
T. W. Korner
Keyword(s):  
Real Variable ◽  
Circular Functions

Mathieu functions of general order: connection formulae, base functions and asymptotic formulae. IV. The Liouville-Green method applied to the Mathieu equation

1981 ◽  
Vol 301 (1459) ◽  
pp. 115-135 ◽  
Keyword(s):  
Full Range ◽  
Mathieu Equation ◽  
Real Variable ◽  
Mathieu Functions ◽  
Asymptotic Formulae ◽  
Half Strip ◽  
Base Functions ◽  
General Order ◽  
Circular Functions ◽  
Connection Formulae

The methods described in part III and the formulae derived in part II are applied to the construction of a comprehensive set of asymptotic formulae relating to the Mathieu equation y ′ ′ + ( λ + 2 h 2 cos ⁡ 2 z ) y = 0 with real parameters. These comprise formulae both ( a ) for the auxiliary parameters and ( b ), in terms of exponential and circular functions, for the fundamental solution, a function of a complex variable, and the various pairs of real-variable base-functions, all introduced in part II. With the aid of these, together with connection formulae also obtained in part II, approximations can readily be obtained for Mathieu functions of various types, including in particular periodic functions. Formulae for solutions are applicable on the half-strip { z : 0 ⩽ Re z ⩽ 1 2 π , Im z ⩾ 0 } , with the transition point of the differential equation which lies on its frontier removed, or in the case of real-variable solutions of the ordinary or modified equation, on the interval [0,½π] or [0, ∞] respectively, with the same qualification as for the half-strip when this is relevant. The formulae cover the full range of the parameters subject to A ≠ ± 2h 2 . The O -terms providing error estimates are uniformly valid on any subdomain of the independent variable and parameters on which they remain bounded.

Real-variable characterizations of Musielak-Orlicz-Hardy spaces associated with Schrödinger operators on domains

2015 ◽  
Vol 39 (3) ◽  
pp. 533-569 ◽  
Author(s):  
Der-Chen Chang ◽  
Zunwei Fu ◽  
Dachun Yang ◽  
Sibei Yang
Keyword(s):  
Hardy Spaces ◽  
Schrödinger Operators ◽  
Real Variable ◽  
Schrodinger Operators

The Dynamic Behaviour of Passively Compensated, Hydrostatic Journal Bearings with Various Numbers of Recesses

1976 ◽  
Vol 18 (6) ◽  
pp. 292-302 ◽  
Author(s):  
P. B. Davies
Keyword(s):  
Dynamic Behaviour ◽  
Damping Ratio ◽  
Equations Of Motion ◽  
Steady State Solution ◽  
Flow Equation ◽  
External Excitation ◽  
Signal Flow Graphs ◽  
Flow Compensation ◽  
Circular Functions ◽  
Flow Graphs

A previously established small-perturbation analysis is developed to express the unsteady-state continuity-of-flow equation for an isolated recess in a passively compensated, multirecess, hydrostatic journal bearing in terms of generalized co-ordinates. The concise form of this equation enables motion of the shaft about the concentric position to be described by equations which are derived in closed form for bearings with orifice, capillary or constant flow compensation and any number of recesses. These equations of motion, and hence the expressions for the receptances which describe the response of a bearing to external excitation, are shown to be of exactly the same form for all bearings of the type considered. Furthermore, the damping ratio and natural frequency in any particular case are determined by a single dynamic constant which is shown to be equal to a linear combination of circular functions and a limited number of coefficients which may be found explicitly by routine use of signal flow graphs. The results of the analysis, which is exact within the stated assumptions, are compared with those of other workers and the steady-state solution of the equations of motion is shown to give an expression for static stiffness which is useful for design purposes. Numerical values of the dynamic constant for bearings with between 3 and 20 recesses are given graphically.

The Theory of Functions of a Real Variable and the Theory of Fourier's Series. Vol. I

1928 ◽  
Vol 14 (192) ◽  
pp. 24
Author(s):  
J. C. Burkill ◽  
E. W. Hobson
Keyword(s):  
Real Variable

scholarly journals Generalised sturm-liouville expansions in series of pairs of related functions

1927 ◽  
Vol 115 (770) ◽  
pp. 184-198 ◽  
Keyword(s):  
Boundary Conditions ◽  
Differentiable Function ◽  
Real Variable ◽  
Complex Parameter ◽  
The Real ◽  
The Earth ◽  
In Series ◽  
Sturm Liouville

The expansions here developed are required for the author’s discussion of "Meteorological Perturbations of Tides and Currents in an Unlimited Channel rotating with the Earth” ( v. supra , p. 170). Let η ( x ) be a real differentiable function of x defined in the range 0 ≼ x ≼ 1, and satisfying the condition η ( x ) > c > 0 for all such x . Let ϕ λ ( x ) and ψ λ ( x ) be functions of the real variable x and the complex parameter λ , defined in the above range by the equations d / dx [ η ( x ) dϕ λ ( x )/ dx ] + ( λ + iγ ) ϕ λ ( x ) = -1, d / dx [ η ( x ) dψ λ ( x )/ dx ] + ( λ + iγ ) ψ λ ( x ) = -1 (1) together with the boundary conditions ϕ' λ (0) = 0, ψ' λ (0) = 0, ϕ' λ (1) = 0, ψ λ (1) = 0, (2) γ being a prescribed constant.

scholarly journals Differentiation in Normed Spaces

2013 ◽  
Vol 21 (2) ◽  
pp. 95-102
Author(s):  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Real Variable ◽  
Normed Spaces

Summary In this article we formalized the Fréchet differentiation. It is defined as a generalization of the differentiation of a real-valued function of a single real variable to more general functions whose domain and range are subsets of normed spaces [14].

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