ON THE SINE INTEGRAL AND THE CONVOLUTION

The sine integral Si(λx) and the cosine integral Ci(λx) and their associated functions Si+(λx) ,S i −(λx) ,C i +(λx) ,C i −(λx) are defined as locally summable functions on the real line. Some convolutions of these functions and sin(µx) ,s in+(µx) ,a nd sin−(µx) are found.

The cosine integral Ci(x) is defined for x > 0 by Ci (see Sneddon [6]).This integral is divergent for x ≤ 0; but in [3], Ci(λx) was defined as a locally summable function on the real line by where H denotes Heaviside's function.In particular, It was proved in [4] that the cosine integral is an even function.We can therefore define Ci(λx) by simplifying the definition given in [3].
The locally summable functions Ci + (λx) and Ci − (λx) are now defined for λ ≠ 0 by It was proved in [3] that where We also need the following lemma which was also proved in [4].
The classical definition of the convolution of two functions f and g is as follows.
Definition 3. Let f and g be functions.Then the convolution f * g is defined by for all points x for which the integral exists.
It follows easily from the definition that if f * g exists then g * f exists and and if (f * g) and f * g (or f * g) exists, then Definition 3 can be extended to define the convolution f * g of two distributions f and g in Ᏸ with the following definition, see Gel'fand and Shilov [5].Definition 4. 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 (15) and ( 16) are satisfied.
Definition 4 of the convolution is rather restrictive and so a neutrix convolution was introduced in [2].In order to define the neutrix convolution we, first of all, let τ be a function in Ᏸ satisfying the following properties: for ν > 0.
The following definition of the neutrix convolution was given in [2].
Definition 8. Let f and g be distributions in Ᏸ and let f ν = f τ ν for ν > 0. Then the neutrix convolution 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 reals, range N the real numbers and with negligible functions finite linear sums of the functions and all functions which converge to zero in the usual sense as ν tends to infinity.
Note that in this definition, the convolution f ν * g is defined in Gel'fand's and Shilov's sense, the distribution f ν having bounded support.
It is easily seen that any results proved with the original definition hold with the new definition.The following theorem (proved in [2]) therefore holds, showing that the neutrix convolution is a generalization of the convolution.
Theorem 9. Let f and g be distributions in Ᏸ satisfying either condition (a) or condition (b) of Definition 4 (Gel'fand's and Shilov's [5]).Then the neutrix convolution f g exists and The next theorem was also proved in [2].
Theorem 10.Let f and g be distributions in Ᏸ and suppose that the neutrix convolution f g exists.Then the neutrix convolution f g exists and Note, however, that the neutrix convolution (f g) is not necessarily equal to f g.We now increase the set of negligible functions given here to include finite linear sums of the functions cos(λν), sin(λν), (λ ≠ 0).

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