Riemann surface of complex logarithm and multiplicative calculus
aa r X i v : . [ m a t h . C V ] O c t Riemann surface of complex logarithm andmultiplicative calculus
Agamirza E. Bashirov
Department of Mathematics, Eastern Mediterranean UniversityGazimagusa, Mersin 10, Turkey andInstitute of Control Systems, ANAS, Baku, Azerbaijan [email protected]
Sajedeh Norozpour
Department of Mathematics, Eastern Mediterranean UniversityGazimagusa, Mersin 10, Turkey [email protected]
Abstract.
Could elementary complex analysis, which covers the topics such as alge-bra of complex numbers, elementary complex functions, complex differentiation andintegration, series expansions of complex functions, residues and singularities, and in-troduction to conformal mappings, be made more elementary? In this paper we demon-strate that a little reorientation of existing elementary complex analysis brings a lot ofbenefits, including operating with single-valued logarithmic and power functions, mak-ing the Cauchy integral formula as a part of fundamental theorem of calculus, removalof residues and singularities, etc. Implicitly, this reorientation consists of resolvingthe multivalued nature of complex logarithm by considering its Riemann surface. Butinstead of the advanced mathematical concepts such as manifolds, differential forms,integration on manifolds, etc, which are necessary for introducing complex analysis inRiemann surfaces, we use rather elementary methods of multiplicative calculus. Wethink that such a reoriented elementary complex analysis could be especially success-ful as a first course in complex analysis for the students of engineering, physics, evenapplied mathematics programs who indeed do not see a second and more advancedcourse in complex analysis. It would be beneficial for the students of pure mathematicsprograms as well because it is more appropriate introduction to complex analysis onRiemann surfaces rather than the existing one.
Keywords.
Complex analysis, complex logarithm, complex exponent, complex dif-ferentiation, complex integration, multiplicative calculus.
AMS Subject Classification.
Primary: 30E20; Secondary: 30E99
Corresponding Author: Agamirza E. Bashirov [email protected] ;Tel: +90 392 6301338; Fax: +90 392 3651604
Elementary complex analysis is a course that is taught for students of pure andapplied mathematics, physics, and engineering programs. This course does notuse the concept of Riemann surfaces and, therefore, presents logarithmic and1ower functions in a multivalued form. This makes elementary complex analysisinsufficiently elementary. An implementation of Riemann surfaces, which treatsthe preceding and other multivalued functions as single-valued, comes later aftergetting knowledges in topology, manifolds, differential forms, integration onmanifolds, etc. In most universities students of applied mathematics, physics,and engineering programs do not see this implementation. Is it possible to makethe elementary part of complex analysis more elementary and at the same timebe closer to the advanced part?This paper demonstrates that if elementary complex analysis is interpretedin the frame of multiplicative calculus, then it turns to be more elementary.In fact, this is a consideration of complex analysis in one particular Riemannsurface (generated by complex logarithm), but does not use heavy machineryof Riemann surfaces. Therefore, multiplicative version of complex analysis alsobecomes a good introductory basis for complex analysis on Riemann surfaces.Elementary complex analysis starts with the field C of complex numbers andoperations on them. Very soon by the de’Moivre’s formula, which states that thenumber of n th roots of any nonzero complex number is exactly n , it becomesclear that the students should expect complications related to this formula.Later this result lies down to the definition of the complex power function thatbecomes multi-valued. Generally, it has countably many branches, it has finitelymany branches just for rational powers that turns to a single branch for integerpowers. Another elementary complex function, met at the beginning, is theargument function arg z with the principal branch Arg z . This function hasagain countably many branches. The most important multivalued function isthe complex logarithmic function log z = ln | z | + i (Arg z + 2 πn ), where | z | is themodulus of z and i is the imaginary unit.Resolving the multivalued nature of these functions without using heavy ma-chinery of Riemann surfaces can essentially simplify elementary complex anal-ysis. To do so, a ”trick” that considers only the principal branches is used.This idea is used in real analysis as well. For example, the familiar sin x ofreal variable has a multi-valued inverse. In order to make it single-valued, wejust invert its restriction to the interval [ − π , π ]. This suffices for the needs ofreal analysis. But for the elementary complex analysis, consideration of onlyprincipal branches is insufficient.The multi-valued nature of the previously mentioned complex functions isdue to their definition over the polar representation of complex numbers which isnon-unique. In fact, there is a lack of space in the complex plane, that containsjust one complex number, corresponding to its distinct polar representations.Introducing the Riemann surface of the complex logarithm solves this insuffi-ciency. But the issue is how to do it for students who are not yet aware abouttopology, differential forms, integration on manifolds, etc. In this paper we aregoing to demonstrate that this issue can be overcame by using easily under-standable concepts of non-Newtonian calculi, specifically, calculus generated byexponential function.The paper is constructed in the following way. In a few following sectionswe discuss some elements towards the basic ideas of the simplification and then2n the concluding section of the paper we propose a draft outline of an alter-native elementary complex analysis. Since the discussion is concentrated onthe elementary part of complex analysis, we call it as complex calculus and thealternative elementary complex analysis as complex *calculus. B We differ distinct polar representations of the same complex number by intro-ducing B = { ( r, θ ) : r > , −∞ < θ < ∞} . (1)Zero will be considered out of B while it belongs to the system C of complexnumbers and the system R of real numbers. This is because the addition op-eration of C as well as R is not fully functioning in B while the multiplicationoperation has a perfect extension. Therefore, it is reasonable to consider B as amultiplicative group rather than a field. One can see that in fact B is the Rie-mann surface of the complex logarithm although this could not be mentionedexplicitly.It is reasonable to call the subset B α = { ( r, θ ) : r > , α − π ≤ θ < α + π } of B as an α -branch of B . In fact B = ∞ [ n = −∞ B πn with B πn ∩ B πm = ∅ for n = m. We will use the identification B ∋ ( r, θ ) = re iθ ∈ C \ { } , where θ stands for the principal argument of the complex number z = re iθ . Inthis regard, C \ { } is a proper subset of B . Similarly, nonzero real numbersbelong to B with representations x = ( x,
0) if x > x = ( − x, − π ) if x < B will be denoted by boldfaceletters like z . Every z = ( r, θ ) ∈ B produces the nonzero complex number z = r (cos θ + i sin θ ). This function will be denoted by z = l ( z ). In fact, thefunction l is the periodic extension from B to B of the preceding identification B = C \ { } . In the sequel, we will use boldface letters like f for functions withdomain or range essentially in B . The symbols like f will be used for functionswith domain and range in C . We will refer to an element from R as r -number,from C as c -number, and from B as b -number for brevity.Not everything is so excellent in B . In particular, the vector addition andsubtraction of c -numbers lose the sense for b -numbers. Instead, the multiplica-tion and division operations are nicely extendable to B as follows z z = ( r , θ )( r , θ ) = ( r r , θ + θ ) (2)3nd z z = ( r , θ )( r , θ ) = ( r /r , θ − θ ) . (3)Therefore, we should avoid addition and subtraction and study functions on B via differentiation and integration on the basis of multiplication and divisionoperations. This can be achieved by implementation of multiplicative definitionsof derivative and integral. Differential and integral calculus, which is briefly called Newtonian calculus,was created by Isaak Newton and Gottfried Wilhelm Leibnitz in the secondhalf of the 17th century. This calculus studies functions via differentiationand integration with reference to linear functions. This means that there is asystem of relationships between certain mathematical concepts, which we nameas CALCULUS (its complex part as COMPLEX CALCULUS), and Newtoniancalculus is its realization in the form of comparison of all functions with linearfunctions. If we choose other reference functions, we could realize CALCULUSdifferently. This issue is similar to a (perfect) translation of a story from onelanguage to another. In the second half of the 60th decade Michael Grossmanand Robert Katz [7] pointed out the realizations of CALCULUS which aredifferent from Newtonian one, calling them as non-Newtonian calculi.Briefly, if α is a bijection from R to the interval I , then addition and multi-plication of R can be isometrically transferred to I by letting a ⊕ α b = α ( α − ( a ) + α − ( b )) and a ⊗ α b = α ( α − ( a ) × α − ( b )) . Similarly, for α -difference and α -ratio we let a ⊖ α b = α ( α − ( a ) − α − ( b )) and a ⊘ α b = α ( α − ( a ) /α − ( b )) . These operations of α -arithmetic simplify many formulae which are less usual inordinary arithmetic. For example, in Einstein’s theory of relativity the speeds a and b are added as a + b ab/c , where c is the speed of light. Let ˜ a = a/c and ˜ b = b/c be the respective relativespeeds. Then (see [3]) a + b ab/c = ˜ a ⊕ tanh ˜ b, where tanh x is the hyperbolic tangent function. This means that formulae intheory of relativity can be more visibly written in terms of tanh-arithmetic.Since the differentiation and integration are infinitesimal versions of subtrac-tion and addition, for a suitable function α , α -arithmetic induces α -derivative4nd α -integral of functions on I as d α dx f ( x ) = α (cid:18) ddx α − ( f ( x )) (cid:19) and ( α ) Z ba f ( x ) dx = α (cid:18) Z ba α − ( f ( x )) dx (cid:19) . The most popular α -calculus is exp-calculus with a reference to exponentialfunction α ( x ) = e x . This calculus is also named as multiplicative calculus, orbriefly *calculus, because exp-addition is the ordinary product: a ⊕ exp b = e ln a +ln b = ab. *Calculus is a suitable calculus for growth related problems. This and otherfeatures of *calculus are discussed in [2, 3]. Numerical methods via *calculusare developed in [6, 8, 9, 10, 11]. Since the addition on B has weaknesses whilethe multiplication is powerful, we can employ multiplicative type derivative andintegral for functions f : B ⊆ B → B .The multiplicative type of concepts are briefly called as *concepts. Previ-ously, an attempt to study *derivative and *integral have been made for func-tions f : C ⊆ C → C in [4, 5]. While *derivative was found satisfactory, *in-tegral met difficulties such as it was defined as a multivalued function becausecomplex log-function is multivalued. Therefore, to go on we need in logarith-mic and exponential functions over the system B , use them to create functions f : B ⊆ B → B , and establish *calculus for them. B On the basis of log z = ln | z | + i (Arg z + 2 πn ) for z ∈ C \ { } , extend log-functionto B as log z = ln r + iθ, z = ( r, θ ) ∈ B , (4)where ln r is the natural logarithm of the r -number r . So, log : B → C becomesa bijection. We define a new exponential function by e z = ( e x , y ) , z = x + iy ∈ C , (5)as a function from C to B . Here e a is a natural exponent of the r -number a (the same symbol will be used for complex exponents in the form e z as wellwhile e z is the new exponential function with values in B ). It can verified that log e z = z and e log z = z for all z ∈ C and z ∈ B . So, they are the inverse ofeach other. These functions play an underlying role in setting calculus over B .With the multiplication operation (2) on B we can establish the followingfamiliar properties of logarithm and exponent. For every z = ( r , θ ) and z = ( r , θ ) from B , log ( z z ) = log z + log z log ( z z ) = log (( r , θ )( r , θ )) = log ( r r , θ + θ ) = ln( r r ) + i ( θ + θ )= ln r + ln r + iθ + iθ = (ln r + iθ ) + (ln r + iθ )= log ( r , θ ) + log ( r , θ ) = log z + log z . Similarly, for every z = x + iy and z = x + iy from C , e z + z = e z e z since e z + z = e ( x + x )+ i ( y + y ) = ( e x + x , y + y )= ( e x e x , y + y ) = ( e x , y )( e x , y ) = e z e z . B The multiplication operation on B induces a natural power of z = ( r, θ ) as z n = ( r, θ ) n = ( r n , nθ ) , and taking into consideration that the r -number 1 has the form (1 ,
0) as a b -number, negative integer power is induced as z − n = ( r, θ ) − n = (1 /r n , − nθ ) , together with z = (1 , ∈ B . The n th root can be defined as a solution of w n = z as w = n √ z = n p ( r, θ ) = ( n √ r, θ/n ) , and hence, the rational power as z p = ( r, θ ) p = ( r p , pθ ) . The complex power w = u + iv of the b -number z = ( r, θ ) can be defined by z w = e w log z = ( e u ln r − vθ , uθ + v ln r ) (6)since z w = e w log z = e ( u + iv )(ln r + iθ ) = e ( u ln r − vθ )+ i ( uθ + v ln r ) = ( e u ln r − vθ , uθ + v ln r )and this agrees with the previously derived formula for rational powers. Thus thepower function f ( w ) = z w for w ∈ C is a single-valued function. In particular, p ( r,
0) = ( √ r,
0) and p ( r, − π ) = ( √ r, − π ) . This means that the square root of a positive r -number r in the sense of B isunique and equals to √ r . Its second root in the sense of R , that is, −√ r isnow the root of ( r, − π ) in the sense of B . Therefore, the positive and negativeroots of a positive r -number are the unique roots of distinct b -numbers whichare identified in R . 6 Other functions
Every single-valued function f of complex calculus generates its analog in com-plex *calculus which we denote by f . The link between f and f is as follows: f ( z ) = e f ( log z ) , z ∈ B ⊆ B . The return link is f ( z ) = log ( f ( e z )) , z ∈ C ⊆ C . In fact, e z , z ∈ C ⊆ B , defined by (5), is the analog of the function e z , z ∈ C . Noting that z ∈ C can be interpreted as c -number z = x + iy = re iθ with x = r cos θ and y = r sin θ as well as b -number z = ( r, θ ), this can be deduces asfollows: e e log z = e e log ( r,θ ) (writing z ∈ C as b -number ( r, θ ))= e e ln r + iθ (definition of log -function)= e r (cos θ + i sin θ ) (definition of exp-function)= ( e r cos θ , r sin θ ) (definition of exp -function)= ( e x , y ) (using x = r cos θ and y = r sin θ )= e z (using z = x + iy )In particular, e z = ( e r cos θ , r sin θ ) for z = re θ ∈ C . This does not expand to multivalued functions such as logarithmic and powerfunctions, which are already defined by (4) and (6), but allows to create analogsof trigonometric and hyperbolic functions. In such a way, cosh z = e ( e log z + e − log z ) = e ( e ln r + iθ + e − ln r − iθ ) = e ( r (cos θ + i sin θ )+ r − (cos θ − i sin θ )) = e r + r − cos θ e i r − r − sin θ = e cosh(ln r ) cos θ e i sinh(ln r ) sin θ = ( e cosh(ln r ) cos θ , sinh(ln r ) sin θ )Similarly, one can derive sinh z = ( e sinh(ln r ) cos θ , cosh(ln r ) sin θ ) , cos z = ( e cosh θ cos(ln r ) , − sinh θ sin(ln r )) , sin z = ( e cosh θ sin(ln r ) , sinh θ cos(ln r )) . *Derivative of functions over B According to Section 3, the *derivative of non-vanishing function f : C ⊂ C → C \ { } is defined by f ∗ ( z ) = e (log f ( z )) ′ = e f ′ ( z ) f ( z ) as a single-valued function [4]. This formula for a function f : R ⊆ R → (0 , ∞ )looks like f ∗ ( x ) = e (ln f ( x )) ′ = e f ′ ( x ) f ( x ) . To extend *derivative to functions f : B ⊆ B → B , we will need the following. Lemma 1.
Let f be a non-vanishing function from some nonempty open con-nected subset C of C to C . Assume that f has the rectangular and polar repre-sentations f ( z ) = u ( r, θ ) + iv ( r, θ ) = R ( r, θ ) e i Θ( r,θ ) for z = re iθ . If f ′ ( z ) exists and, respectively, the Cauchy–Riemann conditions in polar form ru ′ r = v ′ θ , rv ′ r = − u ′ θ hold, then (a) r (ln R ) ′ r = Θ ′ θ and r Θ ′ r = − (ln R ) ′ θ , (b) f ′ ( z ) = e i (Θ − θ ) ( R ′ r + iR Θ ′ r ) , (c) (log f ( z )) ′ = ((ln R ) ′ r cos θ + Θ ′ r sin θ ) + i (Θ ′ r cos θ − (ln R ) ′ r sin θ ) , assumingthat f transfers C into a branch of log-function.Proof. In fact, part (a) expresses a version of the Cauchy–Riemann conditionswhen both arguments and values of a complex function are represented in thepolar form. Most textbooks do not include this version of the Cauchy–Riemannconditions. Therefore, we derive them because b -numbers are based on justpolar representation.We start from R = R ( r, θ ) = p u ( r, θ ) + v ( r, θ ) and calculate R ′ r = uu ′ r + vv ′ r √ u + v = uu ′ r + vv ′ r R and R ′ θ = uu ′ θ + vv ′ θ √ u + v = uu ′ θ + vv ′ θ R .
Next, Θ( r, θ ) = atan2 ( v ( r, θ ) , u ( r, θ )), where the atan2-function is the arctan-function of two variables andatan2 ′ u = − vu + v and atan2 ′ v = uu + v . ′ r = − vu ′ r u + v + uv ′ r u + v = uv ′ r − vu ′ r R . Similarly, Θ ′ θ = − vu ′ θ u + v + uv ′ θ u + v = uv ′ θ − vu ′ θ R . Thus using Cauchy–Riemann conditions in polar form, we obtain rR ′ r = r ( uu ′ r + vv ′ r ) R = uv ′ θ − vu ′ θ R = R Θ ′ θ ⇒ r (ln R ) ′ r = Θ ′ θ and R ′ θ = uu ′ θ + vv ′ θ R = − r ( uv ′ r − vu ′ r ) R = − rR Θ ′ r ⇒ (ln R ) ′ θ = − r Θ ′ r , proving part (a).To prove part (b), we start from the system of equations (cid:26) R ′ r R = uu ′ r + vv ′ r , Θ ′ r R = uv ′ r − vu ′ r . In the matrix form (cid:20) R ′ r R Θ ′ r R (cid:21) = (cid:20) u v − v u (cid:21) (cid:20) u ′ r v ′ r (cid:21) , which implies (cid:20) u ′ r v ′ r (cid:21) = 1 R (cid:20) u − vv u (cid:21) (cid:20) R ′ r R Θ ′ r R (cid:21) . Therefore, u ′ r = 1 R uR ′ r − v Θ ′ r , v ′ r = 1 R vR ′ r + u Θ ′ r . Thus f ′ ( z ) = e − iθ ( u ′ r + iv ′ r ) = e − iθ (cid:18) R ′ r ( u + iv ) R + i Θ ′ r ( u + iv ) (cid:19) = e − iθ ( u + iv ) (cid:18) R ′ r R + i Θ ′ r (cid:19) = e − iθ Re i Θ (cid:18) R ′ r R + i Θ ′ r (cid:19) = e i (Θ − θ ) ( R ′ r + iR Θ ′ r ) , proving part (b).Finally,(log f ( z )) ′ = f ′ ( z ) f ( z ) = e i (Θ − θ ) ( R ′ r + iR Θ ′ r ) Re i Θ = e − iθ ((ln R ) ′ r + i Θ ′ r ) = (cos θ − i sin θ )((ln R ) ′ r + i Θ ′ r )= ((ln R ) ′ r cos θ + Θ ′ r sin θ ) + i (Θ ′ r cos θ − (ln R ) ′ r sin θ ) , proving part (c). 9ow let f : B ⊆ B → B be given. The argument of this function will bedenoted by ( r, θ ) ∈ B ⊆ B and the value by ( R, Θ) ∈ B . Therefore, we canpresent this function as f ( r, θ ) = ( R ( r, θ ) , Θ( r, θ )). For example, for the powerfunction f = z w , z ∈ B , where w = u + iv ∈ C is fixed, the functions R and Θwere derived previously in the form R ( r, θ ) = e u ln r − vθ and Θ( r, θ ) = uθ + v ln r. Motivated from *derivative in the real and complex cases, we will put the for-mula f ∗ ( z ) = e ( log ( f ( z )) ′ on the basis of *derivative of f . Then by Lemma 1(c) we can set the following. Definition 1.
A function f ( r, θ ) = ( R ( r, θ ) , Θ( r, θ )) is said to be *differentiableat ( r, θ ) if R and Θ have continuous partial derivatives and the Cauchy–Riemannconditions in (a) hold at ( r, θ ). If f is *differentiable, then its *derivative isdefined by f ∗ ( z ) = (cid:0) e (ln R ) ′ r cos θ +Θ ′ r sin θ , Θ ′ r cos θ − (ln R ) ′ r sin θ (cid:1) . f is said to be *analytic on B ⊆ B if f is *differentiable at every ( r, θ ) ∈ B . Example 1.
The ordinary derivative of a constant function is equal to theneutral element of addition, that is zero. In real and complex *calculus the*derivative of a constant function is equal to the neutral element of multiplica-tion, that is one. Let us determine whether the *derivative of f ( z ) = z is equalto the neutral element (1 ,
0) of multiplication in B or not. Letting z = ( r, θ ) and z = ( r , θ ), we obtain R ( r, θ ) = r and Θ( r, θ ) = θ . The Cauchy–Riemannconditions (a) hold in the form r (ln R ) ′ r = Θ ′ θ = (ln R ) ′ θ = − r Θ ′ r = 0 . Therefore, f ∗ ( z ) exists. Then by Definition 1, f ∗ ( z ) = ( e ,
0) = (1 , Example 2.
In real and complex *calculus the exponential function plays therole of the linear function and, therefore, ( e z ) ∗ = e (a constant that in realcase shows that the value of this function instantaneously changes e times while( e z ) ′ = e z shows that the value changes for e z units). For e z , we have R ( r, θ ) = e r cos θ and Θ( r, θ ) = r sin θ. The Cauchy–Riemann conditions hold in the form r (ln R ) ′ r = Θ ′ θ = r cos θ and (ln R ) ′ θ = − r Θ ′ r = − r sin θ. We have (ln R ) ′ r cos θ + Θ ′ r sin θ = cos θ + sin θ = 1and Θ ′ r cos θ − (ln R ) ′ r sin θ = sin θ cos θ − cos θ sin θ = 0 , implying ( e z ) ∗ = ( e, xample 3. The *derivative of the identity function f ( z ) = z , z ∈ C \ { } ,was calculated in [5] in the form ( z ) ∗ = e /z . Let us calculate *derivative of f ( z ) = z , z ∈ B . For this function, we have R ( r, θ ) = r and Θ( r, θ ) = θ. Then r (ln R ) ′ r = Θ ′ θ = 1 and (ln R ) ′ θ = − r Θ ′ r = 0 . We have (ln R ) ′ r cos θ + Θ ′ r sin θ = cos θr and Θ ′ r cos θ − (ln R ) ′ r sin θ = − sin θr . Therefore,( z ) ∗ = (cid:18) e r cos θ , − r sin θ (cid:19) = e r (cos θ − i sin θ ) = e r (cos θ + i sin θ ) = e l ( z ) . One can see that ( z ) ∗ is C -valued if − π ≤ − r sin θ < π or, equivalently, − rπ < sin θ ≤ rπ. Example 4.
The *derivative of log z , z ∈ C \ { } , was calculated in [5] in theform (log z ) ∗ = e /z log z as a multi-valued function. Let us calculate *derivativeof log z , z ∈ B . For this function, we have R ( r, θ ) = p ln r + θ and Θ( r, θ ) = atan2 ( θ, ln r ) . Then r (ln R ) ′ r = Θ ′ θ = ln r ln r + θ and (ln R ) ′ θ = − r Θ ′ r = θ ln r + θ . We have (ln R ) ′ r cos θ + Θ ′ r sin θ = ln r cos θ − θ sin θr (ln r + θ )and Θ ′ r cos θ − (ln R ) ′ r sin θ = − θ cos θ + ln r sin θr (ln r + θ ) . Therefore, (log z ) ∗ = (cid:18) e ln r cos θ − θ sin θr (ln2 r + θ , − θ cos θ + ln r sin θr (ln r + θ ) (cid:19) e ln r cos θ − θ sin θ − i (ln r sin θ + θ cos θ ) r (ln2 r + θ = e ln r − iθ ln2 r + θ · cos θ − i sin θr = e log z | log z | · l ( z ) | l ( z ) | = e l ( z ) log z .
11y Cauchy–Schwarz inequality − r p ln r + θ ≤ − θ cos θ + ln r sin θr (ln r + θ ) ≤ r p ln r + θ Therefore, for rπ p ln r + θ >
1, (log z ) ∗ is C -valued. Example 5.
The *derivative of the real power function x a , x >
0, where a ∈ R is fixed, equals to ( x a ) ∗ = e ax . Let us calculate *derivative of z w , z ∈ B ,assuming that w = u + iv ∈ C is fixed. For this function, we have R ( r, θ ) = e u ln r − vθ and Θ( r, θ ) = uθ + v ln r. The Cauchy–Riemann conditions hold in the form r (ln R ) ′ r = Θ ′ θ = u and (ln R ) ′ θ = − r Θ ′ r = − v. We have (ln R ) ′ r cos θ + Θ ′ r sin θ = u cos θ + v sin θr and Θ ′ r cos θ − (ln R ) ′ r sin θ = v cos θ − u sin θr . Therefore, ( z w ) ∗ = ( e ( u cos θ + v sin θ ) /r , ( v cos θ − u sin θ ) /r )= e r ( u cos θ + v sin θ )+ ir ( v cos θ − u sin θ ) . Since w l ( z ) = u + ivr (cos θ + i sin θ ) = ( u + iv )(cos θ − i sin θ ) r = u cos θ + v sin θr + i v cos θ − u sin θr , we obtain ( z w ) ∗ = e w l ( z ) . By Cauchy–Schwarz inequality, −| w | = − p u + v ≤ v cos θ − u sin θ ≤ p u + v = | w | Therefore, for | w | < rπ , ( z w ) ∗ is C -valued.The following properties of *derivative can be easily verified:(i) ( z f ) ∗ ( z ) = f ∗ ( z ) for constant z ∈ B .(ii) ( fg ) ∗ ( z ) = f ∗ ( z ) g ∗ ( z ). 12iii) ( f / g ) ∗ ( z ) = f ∗ ( z ) / g ∗ ( z ).For example, the proof of (ii) is as follows. Letting z = ( r, θ ) , f ( z ) = ( R ( r, θ ) , Θ( r, θ )) , g ( z ) = ( ˜ R ( r, θ ) , ˜Θ( r, θ )) , we have ( fg )( z ) = ( R ( r, θ ) ˜ R ( r, θ ) , Θ( r, θ ) + ˜Θ( r, θ )) . Therefore,( fg ) ∗ ( z ) = (cid:0) e (ln( R ˜ R )) ′ r cos θ +(Θ+ ˜Θ) ′ r sin θ , ((Θ + ˜Θ) ′ r cos θ − ln( R ˜ R )) ′ r sin θ (cid:1) = (cid:0) e (ln R ) ′ r cos θ +Θ ′ r sin θ , Θ ′ r cos θ − (ln R ) ′ r sin θ (cid:1) × (cid:0) e (ln ˜ R ) ′ r cos θ + ˜Θ ′ r sin θ , ˜Θ ′ r cos θ − (ln ˜ R ) ′ r sin θ (cid:1) = f ∗ ( z ) g ∗ ( z ) . The other properties can be proved similarly. Moreover, remaining theorems ofcomplex differentiation can be stated and proved in terms of *derivative. Moreimportantly, we pass to *integral in the next section. B Definition 2.
Given f : B ⊆ B → B by f ( z ) = ( R ( r, θ ) , Θ( r, θ )) for z = ( r, θ ),we assume that C is a contour in B with a parameterization z ( t ) = ( r ( t ) , θ ( t )), a ≤ t ≤ b . Then we define the *integral of f along the curve C by Z C f ( z ) d z = e R C P dr − Q dθ + i R C M dr + N dθ . (7)assuming that the ordinary line integrals in the right side exist and P ( r, θ ) = ln R cos θ − Θ sin θ,Q ( r, θ ) = r ln R sin θ + r Θ cos θ,M ( r, θ ) = ln R sin θ + Θ cos θ,N ( r, θ ) = r ln R cos θ − r Θ sin θ. This integral has the following obvious properties:(i) R C f ( z ) d z = R C f ( z ) d z R C f ( z ) d z , where C = C + C .(ii) R C f ( z ) g ( z ) d z = R C f ( z ) d z R C g ( z ) d z .(iii) R C f ( z ) / g ( z ) d z = R C f ( z ) d z (cid:14) R C g ( z ) d z .For example, property (ii) can be proved as follows. Let P , Q , M , and N bethe preceding functions associated with f and denote the respective functions13ssociated with g by ˜ P , ˜ Q , ˜ M , and ˜ N . Then the respective functions associatedwith fg are P ( r, θ ) = ln( R ˜ R ) cos θ − (Θ + ˜Θ) sin θ, Q ( r, θ ) = r ln( R ˜ R ) sin θ + r (Θ + ˜Θ) cos θ, M ( r, θ ) = ln( R ˜ R ) sin θ + (Θ + ˜Θ) cos θ, N ( r, θ ) = r ln( R ˜ R ) cos θ − r (Θ + ˜Θ) sin θ. Therefore, Z C f ( z ) g ( z ) d z = e R C P dr −Q dθ + i R C M dr + N dθ = e R C P dr − Q dθ + i R C M dr + N dθ e R C ˜ P dr − ˜ Q dθ + i R C ˜ M dr + ˜
N dθ = Z C f ( z ) d z Z C g ( z ) d z . We are mainly interested in whether *integral inverts *differentiation or not.
Theorem 1 (Fundamental theorem of *calculus) . Let f be *analytic on a con-nected set B ⊆ B and let C be a contour in B with the initial and end points z and z , respectively. Then Z C f ∗ ( z ) d z = f ( z ) f ( z ) . Proof.
Let C be parameterized by z ( t ) = ( r ( t ) , θ ( t )), a ≤ t ≤ b , and let f ( z ) =( R ( r, θ ) , Θ( r, θ )). Then f ∗ ( z ) = e (ln R ) ′ r cos θ +Θ ′ r sin θ + i (Θ ′ r cos θ − (ln R ) ′ r sin θ ) . We can calculate the functions P , Q , M , and N , associated with f , in the form: P = ((ln R ) ′ r cos θ + Θ ′ r sin θ ) cos θ − (Θ ′ r cos θ − (ln R ) ′ r sin θ ) sin θ = (ln R ) ′ r ,Q = r ((ln R ) ′ r cos θ + Θ ′ r sin θ ) sin θ + r (Θ ′ r cos θ − (ln R ) ′ r sin θ ) cos θ = r Θ ′ r ,M = ((ln R ) ′ r cos θ + Θ ′ r sin θ ) sin θ + (Θ ′ r cos θ − (ln R ) ′ r sin θ ) cos θ = Θ ′ r ,N = r ((ln R ) ′ r cos θ + Θ ′ r sin θ ) cos θ − r (Θ ′ r cos θ − (ln R ) ′ r sin θ ) sin θ = r (ln R ) ′ r . Therefore, Z C f ∗ ( z ) d z = e R C (ln R ) ′ r dr − r Θ ′ r dθ + i R C Θ ′ r dr + r (ln R ) ′ r dθ .
14y Cauchy–Riemann conditions, Z C f ∗ ( z ) d z = e R C (ln R ) ′ r dr +(ln R ) ′ θ dθ + i R C Θ ′ r dr +Θ ′ θ dθ = e ln R ( r ( b ) ,θ ( b )) − ln R ( r ( a ) ,θ ( a ))+ i (Θ( r ( b ) ,θ ( b )) − Θ( r ( a ) ,θ ( a ))) = e ln R ( r ( b ) ,θ ( b ))+ i Θ( r ( b ) ,θ ( b ))) e ln R ( r ( a ) ,θ ( a ))+ i Θ( r ( a ) ,θ ( a ))) = ( R ( r ( b ) , θ ( b )) , Θ( r ( b ) , θ ( b ))( R ( r ( a ) , θ ( a )) , Θ( r ( a ) , θ ( a )) = f ( z ) f ( z ) , proving the theorem. Remark 1.
Let C and C be two contours in the connected set B ⊆ B withthe same initial and end points z and z , respectively, and let f be *analyticon B . Then by Theorem 1, Z C f ( z ) d z = Z C f ( z ) d z , that means the *integral is independent on the shape of the contours. Therefore,this integral can be denoted as Z z z f ( z ) d z . Theorem 2.
Let f be *analytic on a connected set B ⊆ B and let C be a closedcontour in B . Then Z C f ∗ ( z ) d z = (1 ,
0) = e . Proof.
Under notation in Definition 2, we have Q ′ r + P ′ θ = ln R sin θ + Θ cos θ − ln R sin θ − Θ cos θ = 0and N ′ r − M ′ θ = ln R cos θ + Θ sin θ − ln R cos θ − Θ sin θ = 0 . Therefore, applying Green’s theorem to (7), we obtain the conclusion of thetheorem.
Example 6.
By Example 3, ( z ) ∗ = e l ( z ) . We are seeking to find the analog ofthis formula in complex calculus. In the real case, this formula states e x = f ∗ ( x ) = e f ′ ( x ) f ( x ) ⇒ (ln f ( x )) ′ = 1 x ⇒ f ( x ) = x + c. Therefore, the equality ( z ) ∗ = e l ( z ) is a complex *version of ( x ) ′ = 1. Thenwhat is the analog of I | z | = a z dz = 015n complex *calculus? The analog of the counterclockwise oriented circle | z | = a in C is the contour C in B with the parameterization ( r, θ ) = ( a, t ), − π ≤ t ≤ π ,which is no longer closed in B . It has distinct initial and end points ( a, − π )and ( a, π ), respectively, in disjoint branches. On the basis of Theorem 1 andExample 3, we can calculate Z C (cid:0) e l ( z ) (cid:1) d z = z z = ( a, π )( a, − π ) = (1 , π ) = e πi . While expecting to obtain the b -number (1 ,
0) (e.g. c -number 1 that is a neutralelement of multiplication), we obtained (1 , π ). This is because in C the b -numbers (1 ,
0) and (1 , π ) are identified. Example 7.
By Example 4, ( log z ) ∗ = e l ( z ) log z . In the real case, this formulastates e x ln x = f ∗ ( x ) = e f ′ ( x ) f ( x ) ⇒ (ln f ( x )) ′ = 1 x ln x ⇒ f ( x ) = ln x + c. Therefore, the equality ( log z ) ∗ = e l ( z ) log z is a complex *version of (ln x ) ′ = x .Based on this we would like to find *version of the important formula I | z | = a dzz = 2 πi for a > . Similar to Example 6, we look to the contour C in B with parameterization( r, θ ) = ( a, t ), − π ≤ t ≤ π . On the basis of Theorem 1 and Example 4, we have Z C (cid:0) e l ( z ) log z (cid:1) d z = log ( a, π ) log ( a, − π ) = e i arctan π ln a . The dependence on a of the calculated result looks like strange, but this isaccumulated from *nature of calculations. Indeed, converting the result fromexp-arithmetic to ordinary arithmetic by formula z − z = log ( e z ⊖ exp e z ),we obtain exactly 2 πi : log (cid:0) e log ( a,π ) ⊖ exp e log ( a, − π ) (cid:1) = log e ln a + iπ − ln a + iπ = log ( e , π ) = ln 1 + 2 πi = 2 πi. This example demonstrates that the *integral responds to residues in a specificform. More precisely, the residues appear as a result of calculation of *integralsover non-closed contours by using the fundamental theorem of *calculus. Inother words, the residue theory, which is a some sort of attachment to funda-mental theorem of complex calculus and covers the cases out of this theorem,now becomes an integral part of fundamental theorem of complex *calculus.16
Conclusion
On the basis of the preceding discussion we find it reasonable to develop othertopics of elementary complex calculus in multiplicative frame in details with theaim to propose an alternative course on elementary complex calculus. The draftoutline of this course may be seen as follows: • Definition of C and algebra on C including the de’Moivre’s formula. TheEuler’s formula, periodic exponential function, multivalued logarithmicand power functions. Complications related to multivalued functions. • Definition of B without mentioning that it is the Riemann surface of log-function. Algebra on B emphasizing on global nature of multiplication on B . • Review on non-Newtonian calculi in the real case emphasizing on *calculus.Pointing out that *calculus is a presentation of CALCULUS and it isequivalent to Newtonian presentation. • Definitions of elementary functions (logarithmic, exponential, power, trigono-metric, hyperbolic) over B . • Limit and continuity of f ( z ) = ( R ( r, θ ) , Θ( r, θ )). Although this part was notconcerned in the preceding discussion, it can be developed on the basisof limit and continuity of functions R and Θ of two real variables. It issuitable to discuss limits at zero and infinity. • *Differentiation with Cauchy–Riemann conditions from Lemma 1(a). The*derivative of f ( z ) = ( R ( r, θ ) , Θ( r, θ )) can be developed by using *deriva-tives of real-valued function R and Θ through limit and then Definition 1can be obtained. • *Integration with fundamental theorem of *calculus. Again *integrals canbe developed by using of integral products (instead of integral sums) andthen Definition 2 can be obtained. • No need for residues and singularities in complex *calculus since complex*integrals can be calculated just by fundamental theorem of *calculus.Also, the functions of complex *calculus have no singularities leading toresidues. Removal of this part of complex calculus would provide a greateconomy of time which can be used for next items. • Conformal mappings. This can be developed in *context in more detailsbecause of saving time. The preceding discussion does not include thistopic which needs additional investigation. • In addition, working on new topics in *context is recommended. For example,complex Fourier series in *context seems to be appropriate. The precedingdiscussion does not include this topic. It is notable that Fourier series17t to Newtonian calculus inappropriately in comparison to Taylor seriesbecause the reference function of Newtonian calculus is linear. Fourierseries do not fit to real *calculus as well because sine and cosine functionshave zeros. We think that an appropriate calculus for Fourier series iscomplex *calculus over B because its reference function is non-vanishingexponential function e z and also complex Fourier series have exponentialform. This issue needs additional investigation.Of course, there are a lot of problems related to interpretation of remainingtheorems of complex analysis (elementary or not) in *form. Besides conformalmappings and Fourier series, mentioned previously, it is challenging a study ofthe the fundamental theorem of algebra in *context. Noting that in *calculuspolynomials become products of power functions and motivated from the single-valued nature of power functions in complex *calculus, fundamental *theoremof algebra is expecting to be seen as an existence of a unique solution.This paper just demonstrates that the Riemann surface of complex loga-rithm can be replaced by methods of multiplicative calculus. This raises thefollowing challenging question: Can complex analysis on Riemann surfaces beeasily covered by consideration different non-Newtonian calculi? This requiresa wide research of interested experts in the field. We expect many items ofcomplex analysis to be changed, mainly being simplified. References [1] Ahlfors, L. V.: Complex Analysis, 3rd ed. McGraw-Hill, New York (1979)[2] Bashirov, A. E., Kurpınar, E., ¨Ozyapıcı, A.: Multiplicative calculus and itsapplications. Journal of Mathematical Analysis and Applications. (1) ,36–48 (2008)[3] Bashirov, A. 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