| P-adic numbers were first introduced in 1898 by the German mathematician K. Hensel in a paper which was concerned with the development of algebraic numbers in power series. In the beginning, they were used mainly in number theory. Since 80th p-adic numbers were used in applications to quantum physics. P-adic strings and superstrings were the first models of p-adic quantum physics. The interest of physicists to p-adic numbers is explained by the attempts to create new models of space-time for the description of Planck distances (the distance is no more than 1.6 × 10-33m, time interval is within 5.4 × 10-44s). There are some evidences that the standard model based on real numbers is not adequate to Planck's domain. On the other hand, some properties of fields of p-adic numbers seem to be closely related to Plank's domain, (see [A.D.F.V], [V2])The pioneer investigations on p-adic string theory induced investigations on p-adic quantum mechanics and field theory. These investigations induce a development of p-adic mathematics in many directions: theory of differential and pseudodifferential equations (see [Khr1], [V2], [WZ1], [Koc1]-[Koc4]), theory of probability (see [A.C.D.K], [Khr2], [Khr3], [WZ1], [V2]) and so on. These applications induce numerous new mathematical problems. Then p-adic numbers were generalized to valuations by Kurschak in 1913. In 1923 Hasse subsequently formulated the local-global principle (now usually called the Hasse principle), which is one of the chief applications of local field theory. Skolem's p-adic method, which is used in attacking certain Diophantine equations, is another powerful application of p-adic numbers.There are two major kind of functions based on p-adic numbers. One is from Qp to Qp (see [C], [Kob2], [Kob1], [S]). The other is from Qp to C (see [Koc1]-[Koc5], [T], [Vo]). In this paper, we mainly consider the second one. In [V1], defined the general function's derivative formulas by... |
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