EXPANSION OF A CLASS OF FUNCTIONS INTO AN INTEGRAL INVOLVING ASSOCIATED LEGENDRE FUNCTIONS NANIGOPAL

A theorem for expansion of a class of functions into an integral involving associated Legendre functions is obtained in this paper. This is a soxnewhat general integral expansion formula for a function f(z) defined in (Zl,Z2) where < x < 2 < l, which is perhaps useful in solving certain boundary value problems of mathematical physics and of elasticity involving conical boundaries.

Integral transforms are often used to solve the problems of mathematical physics involving linear partial differential equations and also other problems.Integral expansions involving spherical functions of a class of functions are known as Mehler-Fok type transforms.In these transform formulae, the subscript of the Legendre functions appear as the integration variable while its superscript is either zero or a fixed integer (see Sneddon  [10]).There is another class of integral transforms involving associated Legendre functions somewhat related to the Mehler-Fok transforms, in which the superscript of the associated Legendre.function appears in the integration formula while the subscript (complex) is kept fixed.Felsen [2] first developed this type of transform formulae involving P-/2 + ir (cs O) as kernel where 0 < 0 < r from a unique 6- function representation.Later Mandal ([6], [7]) obtained somewhat similar types of two transform formulae from the solution of two appropriately designed boundary value problems.In the first type, the argument z of P-/2 + ir (z) ranges from -1 to while in the second, the argument z of P_ 1/2 +it (z) ranges from to oo.Recently Mandal and Guha Roy [8] used a similar technique to establish another Mehler-Fok type integral transform formula involving P [ /2 + ir (cs O) as kernel (0 < 0 < a).
In the present paper, an integral expansion of a class of functions defined in (Zl,Z2) where -1 <rl <z2<l, involving associated Legendre functions is obtained.Based on direct investigation of the properties of spherical functions, sufficient conditions which would establish the validity of this expansion formula for a wide class of functions are obtained in a manner similar to the ideas used in ([3]- [5]).The main result is given in section 2 in the form of a theorem.Recently, we have used a similar technique to establish another type of integral representation [9] involving P-/2 + ir (esh ) as kernel where 0 < c, < a0" We present the main result of this paper in the form of the following theorem.
THEOREM.Let f(z) be a given function defined on the interval (Zl,Z2) where < z < z 2 < and satisfies the following conditions: (1) The function f(z) is piecewise continuous and has a bounded variation in the open interval (Zl, z2).
It follows from the properties of associated Legendre functions (cf.Erd61yi [1]) that the integrand in the above integral is an odd function of o', hence the integral vanishes.Thus (2.6) To investigate the behavior of K(,v,T) as T.--.oo, by writing # -ir, we write (2.5) (2.7)   iT Expression under the integral sign in (2.7) is analytic function fo the complex variable # and it has no singularity in the semi-plane Re# >_ O, except for simple poles at # -io" k (k is positive integer) (cf.Felsen  [2]), where M(z2, zl;io'k) 0, o'k > 0. (2.8) Completing the contour of integration on (2.7) with the arc IT of radius T situated in the semi-plane Re# > 0 and applying the residue theorem, we obtain e(z, z2; io'k)m(v, z 1; io" k) K(z'!#'T)=KI("r'!#'T)-E(2.9) k where Suppose that v < .By virtue of the definition [l+o(iui-1)) P r(l+u) [1 + o( I,. l.)]
Moreover, if the integral of integration is divided into the subintervals (-,) and (a,-) and if a sufficiently small positive (implying a sufficiently large T) is chosen, then we have lY(tanh ')1 (2.17) Thus, at the points of continuity of f(z) we obtain (2.1).We note that (2.1) becomes result in [5] when z -1 and z 2 1.
It follows from the foregoing theorem that, at points of continuity of y(z), we have and r a k s,a, are real.
The integrand in (2.21) has singularities at a=ak(k is positive integers) which are simple poles along the positive a-axis, where 0 R(Z, zl;iak O, > 0 o-- To prove (2.21) we use the following asymptotic formulas for large 0 P-f/2 it(')--P (_)-#/2 O( i-l)], o- o,,: The proof of (2.21) is similar to the proof in the section 2, and we do not reproduce it.We note that (2.21) becomes a result in [5] when 3. EXAMPLES.We now give examples of expansions of some functions.
In M1 these results the conditiom under which the expsion theorem hold e satisfied.
ACKNOWLEDGEMENT.This research is supported by CSIR, New Delhi, through a research project No. 25(41)/EMR-II/88 administered by the Calcutta Mathematical Society.

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Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
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