Plemelj Formula of Cauchy-Type Integral of Random Process with Second Order Moment

The research is Supported by Natural Science Foundation of Fujian Province, China (2007J0183). Abstract Under the condition of arc-wise smooth path of integration, the Plemelj formula of Cauchy-type integral on random process with second order moment is obtained.


Introduction
Let L be a simple, smooth and closed curve.It divides the complex plan into inter domain D + and outer domain D − .(Ω, F, P) is a probability space, g(ω, ζ) is a random process with second order moment on (Ω, F, P), which depends on parameters ζ on L. In [Wang, 2004], Wang Chuangrong gave the definition of random cauchy-type integral of g(ω, ζ), and proved the existence of random singular integral on arc-wise smooth curve L. In [Wang, 2005], the author discussed some properties of random singular integral, proved that random singular integral operator was a linear bounded operator and gave the plemelj fomula of random cauchy-type integral on smooth curve L. In this paper, we continue to consider random singular integral of random process with second order moment, and we get the plemelj formula of general form when L is a an arc-wise smooth curve.It is well known that singular integral equation and boundary value problems of analytic function and random process are closely connected with many physical and engineering problems such as elastic mechanics, crack mechanics and aero-dynamics, ect.Therefore it is expected that the results of present paper will be applied in future.

Some Preliminaries
Lemma 1 Let L be an arc-wise smooth and closed curve, ) can be correspondingly understood by g(τ, z + ) and g(τ, z − ).
Proof We only proof that g(τ, z) ∈ H when τ ∈ L, z ∈ D + .So it is sufficient to prove that: g(τ, z), as a function of one of its arguments, ∈ H uniformly with respect to the other argument.For any t 1 , t 2 , τ 1 , τ 2 ∈ L, we have At first, if τ is fixed, we need to prove the following inequality holds for any z 1 , z 2 ∈ D + , where A 1 is independent of τ.According to the proof of Privalov theorem [Lu, Jianke, 2004], we can know A 1 is really independent of τ.
Secondly, if z is fixed, we need to prove the following inequality holds for any τ 1 , τ 2 ∈ L, where β 1 is independent of z, it can be proved by the proof of corollary 2 of theorem 1.3 [Hou Zongyi, 1990].
Lemma 2 Let L be an arc-wise smooth closed curve and f (t, z) ∈ H when t ∈ L, z ∈ T , where T is a region containing L in its interior.Let Then for any t ∈ L, we have where θ 0 is the angle spanned by the two one-sided tangents at ζ 0 towards the positive side of L. If f (t, z) is defined and fulfills the assumed condition only for z in one side of L (including L itself), then the conclusion is also valid for the boundary value of the same side.
The above lemma can be proved by the proof of theorem 1.4.2[Lu Jianke, 2004].
Lemma 3 Let L be an arc-wise smooth and closed curve, then g(τ, t 0 ), as a function of τ, ∈ H uniformly with respect to t 0 , which can be proved by the proof of corollary 2 of theorem 1.3 [Hou Zongyi, 1990].
Lemma 4 Let L be an arc-wise smooth and closed curve, then we get thus, we have By lemma 1 and lemma 2, we get According to the plemelj fomula, we have

Main Results
Let we have and Then the Plemelj formula holds in the sense of mean square metric, where where θ 0 is the angle spanned by the two one-sided tangents at ζ 0 towards the positive side of L. And By lemma 4, we have lim And according to lemma 3 and Plemelj formula, we can get Similarly, we have Analogously, we can prove that lim The proof is complete.