INTEGRALS OF A REAL VARIABLE

1974 ◽  
pp. 66-105
Author(s):  
F.W.J. Olver
Keyword(s):  
Real Variable

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].

SPHERICAL FUNCTIONS, THE COMPLEX HYPERBOLIC PLANE AND THE HYPERGEOMETRIC OPERATOR

2006 ◽  
Vol 17 (10) ◽  
pp. 1151-1173 ◽  
Author(s):  
P. ROMÁN ◽  
J. TIRAO
Keyword(s):  
Hypergeometric Function ◽  
Hyperbolic Plane ◽  
Casimir Operator ◽  
Differential Operators ◽  
Real Variable ◽  
Spherical Functions ◽  
Matrix Coefficients ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Complex Hyperbolic Plane

In this paper, we determine all irreducible spherical functions Φ of any K-type associated to the dual Hermitian symmetric pairs (G, K) = ( SU (3), U (2)) and ( SU (2,1), U (2)). This is accomplished by associating to Φ a vector valued function H = H(u) of a real variable u, analytic at u = 0, which is a simultaneous eigenfunction of two second order differential operators with matrix coefficients. One of them comes from the Casimir operator of G and we prove that it is conjugated to a hypergeometric operator, allowing us to express the function H in terms of a matrix valued hypergeometric function. For the compact pair ( SU (3), U (2)), this project was started in [4].

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