View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Computers and Mathematics with Applications 60 (2010) 2725–2737 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Multiplicative type complex calculus as an alternative to the classical calculus Ali Uzer ∗ Department of Electrical and Electronics Engineering, Fatih University, Istanbul, Turkey article info a b s t r a c t Article history: A multiplicative calculus dealing with real valued functions is extended to a multiplicative Received 21 April 2009 type complex calculus (MCC) dealing with complex valued functions. Some fundamental Received in revised form 29 August 2010 theorems and concepts of the classical calculus are interpreted from the view point of Accepted 30 August 2010 the MCC and the analogies between them are given. Also new notations for the MCC are defined. The MCC is distinguished from the classical calculus by calling the classical calculus Keywords: as the additive type complex calculus (ACC). Benford effect ' 2010 Elsevier Ltd. All rights reserved. Multiplicative calculus Non-Newtonian calculus Change rate 1. Introduction Many of the scientific tables and graphs are given in logarithmic scales. For example, the levels of sound signals, the acidities of chemicals, and the magnitudes of earthquakes are all given in logarithmic scales. In fact, an important work of Benford [1] that was published 70 years ago implies many physical quantities in the nature are of exponentially varying type. In his work, Benford made a statistical work on twenty different tables containing about 20,000 quantities, which had been taken from the real world. He observed that the quantities such as the surface areas of rivers, the population figures of countries, the electricity bills, the stock prices in markets, etc. are all of exponentially varying type. Extensive discussions on such quantities can be found in [2]. The multiplicative calculus, which seems convenient for dealing with exponentially varying functions, was firstly proposed by mathematical biologists Volterra and Hostinsky [3] in 1938. In the following decades, the multiplicative calculus was considered in some other books and papers. A few of the very important ones are [4–7]. Notations of the multiplicative ∗ d calculus in the existing literature are as follows: the multiplicative derivative of a function f .x/ is denoted as dx∗ f .x/ or sometimes as f ∗.x/. Its explicit definition is given by d∗ f .x C h/1=h VD I 2 C ∗ f .x/ lim f .x/ R ; (1) dx h!0 f .x/ C R b dx Qb dx where R denotes the positive real numbers. The multiplicative integral is denoted as a f .x/ or sometimes as a f .x/ and is defined as Z b Z b dx C f .x/ VD exp ln f .x/dx I f .x/ 2 R ; a a ∗ Tel.: +90 212 866 33 00x5586; fax: +90 212 8663412. E-mail address: [email protected]. 0898-1221/$ – see front matter ' 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2010.08.089 2726 A. Uzer / Computers and Mathematics with Applications 60 (2010) 2725–2737 where ln./ means the natural logarithm function. The asterisk .∗/ in (1) is used for distinguishing the multiplicative derivative from the classical derivative. The two derivatives are related by ∗ d d C f .x/ D exp ln f .x/ I f .x/ 2 R : dx∗ dx Our search on the literature reveals that the multiplicative calculus deals only with real valued functions. In this paper we extend it for dealing with complex valued functions. The new calculus will be then called the multiplicative type complex calculus (MCC). Whilst the classical calculus will be called the additive type complex calculus (ACC). Examples are given to show some analogies between them. 2. Definitions, methods, and theorems 2.1. A multiplicative group, an additive group, and an isomorphism Consider two sets of complex numbers Upun and Ustr , which are given by Upun VD C n f0g; (2) Ustr VD fu C iv V u 2 RI − π ≤ v < πg: (3) Assume a one-to-one and onto mapping L V Upun 7! Ustr is defined as p Lfx C iyg VD lnf x2 C y2g C iArgfx C iyg; (4) − L 1fu C ivg VD exp.u C iv/; where ln./ means the natural logarithm function and Argfx C iyg means the polar angle of the number x C iy such that −π ≤ Argfx C iyg < π: In fact the operator L in (4) is equivalent to the principle value of the complex logarithm function, L ≡ Log; − L 1 ≡ exp : We will use in this paper the notations L and L−1 in order to avoid lengthy statements in equation writings. We will specify group operators relevant to the sets Upun and Ustr given in (2) and (3). Let the set Upun be equipped with the ordinary multiplication operator .·/. Then hUpun; ·⟩ can be called a group since it satisfies the group axioms: • Closure in Upun: z1z2 2 Upun for all z1; z2 2 Upun. • Associativity in Upun: .z1z2/z3 D z1.z2z3/ for all z1; z2; z3 2 Upun. • Identity in Upun: There is an element 1 2 Upun such that z1 D 1z D z for all z 2 Upun. • Inverse in Upun: For every z 2 Upun, there exists one-and-only-one inverse element inv.z/ 2 Upun such that z .inv.z// D .inv.z// z D 1. Indeed inv.z/ D 1=z. Next assume the set Ustr is equipped with an operator ⊕ that is defined by any of the following expressions: −1 w1 ⊕ w2 VD L L fw1 C w2g ; =w1 C =w2 (5) w1 ⊕ w2 VD w1 C w2 − i2πNint ; 2π where the function Nint./ rounds its argument value to a nearest integer. Then hUstr ; ⊕⟩ can be called a group since it satisfies the axioms: • Closure in Ustr : w1 ⊕ w2 2 Ustr for all w1; w2 2 Ustr . • Associativity in Ustr : .w1 ⊕ w2/ ⊕ w3 D w1 ⊕ .w2 ⊕ w3/ for all w1; w2; w3 2 Ustr . • Identity in Ustr : There is an element 0 2 Ustr such that w ⊕ 0 D 0 ⊕ w D w for all w 2 Ustr . • Inverse in Ustr : For every w 2 Ustr , there exists one-and-only-one inverse element inv.w/ 2 Ustr such that w ⊕ inv.w/ D inv.w/ ⊕ w D 0. The inverse of an element can be found from any of − inv.w/ D LfL 1{−wgg; =w inv.w/ D −w C i2πNint ; 2π inv.w/ D 0 ⊕ .−w/: As a final property, the mapping L forms an isomorphism between the groups hUpun; ·⟩ and hUstr ; ⊕⟩. That is, the following three axioms are satisfied A. Uzer / Computers and Mathematics with Applications 60 (2010) 2725–2737 2727 • L is one-to-one and onto. • The group operation .·/ in hUpun; ·⟩ corresponds to the group operation ⊕ in hUstr ; ⊕⟩. That is, Lfz1z2g D Lfz1g ⊕ Lfz2g for all z1; z2 2 Upun. • The image of the identity element 1 2 Upun is the identity element 0 2 Ustr . That is, Lf1g D 0. ∼ This isomorphism is denoted as hUpun; ·⟩ D hUstr ; ⊕⟩. It can be used to develop the MCC as we do in the following section. 2.2. Remoteness of two values The remoteness of any two complex quantities za,zb can be defined in the sense of the ACC as rACC .za; zb/ VD zb − za: (6) However, in the sense of the MCC, the numbers za; zb can be interpreted as two elements in the group hUpun; ·⟩. By mapping the elements into the group hUstr ; ⊕⟩, the remoteness may be measured as Lfzbg ⊕ .−Lfzag/. Then the inverse mapping of this expression back into the group hUpun; ·⟩ will give zb=za. Hence the remoteness of any two complex quantities za; zb can also be defined as zb rMCC .za; zb/ VD : (7) za Both definitions in (6) and (7) are meaningful to measure the remoteness of complex numbers and are said to be analogous to each other. Particularly, definition (7) will be used to develop the MCC. Example 1 (Simple Interpolations). Let the two values of the Bessel function J5.z/ be available on a table as fa D J5.za/ ≈ −2:3179 − 0:4077i; fb D J5.zb/ ≈ −1:9456 C 0:3870i; at the arguments −iπ=12 za D −9 C 4i C 2e ; −iπ=12 zb D −9 C 4i C 3e ; and the function value is needed at an intermediate argument value. A simple interpolating equation in the sense of the ACC can be set up as follows: z − za fadd.z/ D fa C .fb − fa/ ≈ −2:3179 − 0:4077i C .0:5618 C 1:4636i/.z − za/: zb − za However, the interpolation can also be done in the sense of the MCC. The values fa and fb can be interpreted as two elements in the group hUpun; ·⟩. After mapping those elements into the group hUstr ; ⊕⟩, an interpolating equation can be set up as z − za Lffag ⊕ .Lffbg ⊕ .−Lffag// : zb − za Then by mapping this expression back into the group hUpun; ·⟩, an interpolating equation will be obtained z−za − f zb za b .−0:1607−0:7951i/.z−za/ fmult .z/ D fa ≈ .−2:3179 − 0:4077i/e : fa For testing the accuracies, the functions fadd.z/, fmult .z/, and J5.z/ are computed on a contour − γ VD z V z D −9 C 4i C te iπ=12I 0 ≤ t ≤ 20 ; and the values are plotted as the real parts versus the imaginary parts in Fig. 1. The function fmult .z/ is seen to be a better approximation than the function fadd.z/.