© Hindawi Publishing Corp. ON THE FRESNEL SINE INTEGRAL AND THE CONVOLUTION

The Fresnel sine integral S(x), the Fresnel cosine integral C(x), and the associated functions S


Introduction.
The Fresnel integrals occur in the diffraction theory and they are of two kinds: the Fresnel integral S(x) with a sine in the integral and the Fresnel integral C(x) with a cosine in the integral.
The Fresnel sine integral S(x) is defined by x 0 sin u 2 du (1.1) (see [5]) and the associated functions S + (x) and S − (x) are defined by S + (x) = H(x)S(x), S − (x) = H(−x)S(x). (1. 2) The Fresnel cosine integral C(x) is defined by (see [5]) and the associated functions C + (x) and C − (x) are defined by where H denotes Heaviside's function.
We define the function L r (x) by L r (x) = x 0 u r sin u 2 du (1.5) for r = 0, 1, 2,....In particular, we have (1.6) We define the functions sin + x, sin − x, cos + x, and cos − x by 2. Convolution products.The classical definition for the convolution product of two functions f and g is as follows.
Definition 2.1.Let f and g be functions.Then the convolution f * g is defined by for all points x for which the integral exists.
If the classical convolution f * g of two functions f and g exists, then g * f exists and (2.2) Further, if (f * g) and f * g (or f * g) exist, then The classical definition of the convolution can be extended to define the convolution f * g of two distributions f and g in Ᏸ with the following definition, see [4].
Definition 2.2.Let f and g be distributions in Ᏸ .Then the convolution f * g is defined by the equation for arbitrary ϕ in Ᏸ , provided that f and g satisfy either of the following conditions: (a) either f or g has bounded support, (b) the supports of f and g are bounded on the same side.
It follows that if the convolution f * g exists by this definition, then (2.2) and (2.3) are satisfied.

Proof.
It is obvious that thus proving (2.5).

Existence of neutrix convolution product.
In order to extend the convolution product to a larger class of distributions, the neutrix convolution product was introduced in [1] and was later extended in [2,3].For the further extension, first of all, we let τ be a function in Ᏸ having the following properties: Definition 3.1.Let f and g be distributions in Ᏸ and let f ν = f τ ν for ν > 0. The neutrix convolution product f g is defined as the neutrix limit of the sequence {f ν * g}, provided that the limit h exists in the sense that for all ϕ in Ᏸ, where N is the neutrix, see van der Corput [7], having domain N , the positive real numbers, with negligible functions finite linear sums of the functions ν λ ln r −1 ν, ln r ν, ν r sin ν 2 , and ν r sin ν 2 (λ = 0, r = 1, 2,...) and all functions which converge to zero in the normal sense as ν tends to infinity.
Note that in this definition the convolution product f ν * g is defined in Gel'fand and Shilov's sense, with the distribution f ν having bounded support.
It was proved in [1] that if f * g exists in the classical sense or by Definition 2.1, then f g exists and The following theorem was also proved in [1].
Theorem 3.2.Let f and g be distributions in Ᏸ and suppose that the neutrix convolution product f g exists.Then the neutrix convolution product f g exists and Now if we let L r = N-lim ν→∞ L r (ν) and note that see Olver [6], then we have the following theorem.
Then the convolution (sin + x 2 ) ν * x r exists and and it follows that Further, it can easily be seen that for each fixed x, and (3.6) follows from (3.9), (3.10), and (3.11).

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation