Bigeometric Calculus and its applications
aa r X i v : . [ m a t h . G M ] A ug Bigeometric Calculus and its applications
Khirod Boruah and Bipan Hazarika ∗ Department of Mathematics, Rajiv Gandhi University, Rono Hills,Doimukh-791112, Arunachal Pradesh, IndiaEmail: [email protected]; bh [email protected]
Abstract.
Based on M. Grossman in [10] and Grossman an Katz [9], in this paper wediscuss about the applications of bigeometric calculus in different branches of mathemat-ics and economics.
Keywords and phrases:
Geometric real numbers; geometric arithmetic; bigeometric-derivative; bigeometric-continuity.
AMS subject classification (2000): Introduction
In the area of non-Newtonian calculus pioneering work carried out by Grossman andKatz [9] which we call as multiplicative calculus. The operations of multiplicative calculusare called as multiplicative derivative and multiplicative integral. We refer to Grossmanand Katz [9], Stanley [16], Campbell [7], Bashirov et al. [2, 3], Grossman [10, 11], JaneGrossman [12, 13] for different types of Non-Newtonian calculus and its applications.An extension of multiplicative calculus to functions of complex variables is handled inBashirov and Rıza [1], Uzer [19], Bashirov et al. [3], C¸ akmak and Ba¸sar [6], Tekin andBa¸sar[17], T¨urkmen and Ba¸sar [18]. The generalized Runge-Kutta method with respectto non-Newtonian calculus studied by Kadak and ¨Ozl¨uk [14].Bigeometric-calculus is an alternative to the usual calculus of Newton and Leibniz.It provides differentiation and integration tools based on multiplication instead of addi-tion. Every property in Newtonian calculus has an analog in Bigeometric-calculus. Gen-erally, in growth related problems, price elasticity, numerical approximations problemsBigeometric-calculus can be advocated instead of a traditional Newtonian one.Throughout the article for our convenience we will used ” G -Calculus” instead of ”Bigeometric-Calculus”. 2. α -generator and geometric real field A generator is a one-to-one function whose domain is R (the set of real numbers) andwhose range is a subset B ⊂ R . Each generator generates exactly one arithmetic andeach arithmetic is generated by exactly one generator. For example, the identity functiongenerates classical arithmetic, and exponential function generates geometric arithmetic. ∗ The corresponding author.September 1, 2016.
As a generator, we choose the function α such that whose basic algebraic operations aredefined as follows: α − addition x ˙+ y = α [ α − ( x ) + α − ( y )] α − subtraction x ˙ − y = α [ α − ( x ) − α − ( y )] α − multiplication x ˙ × y = α [ α − ( x ) × α − ( y )] α − division ˙ x/y = α [ α − ( x ) /α − ( y )] α − order x ˙
For all x, y ∈ R ( G ) • x ⊕ y = xy • x ⊖ y = x/y • x ⊙ y = x ln y = y ln x • x ⊘ y or xy G = x y , y = 1 • x G = x ⊙ x = x ln x • x p G = x ln p − x • √ x G = e (ln x ) • x − G = e x • x ⊙ e = x and x ⊕ x • e n ⊙ x = x ⊕ x ⊕ ..... (upto n number of x ) = x n • | x | G = x, if x > , if x = 1 x , if 0 < x < | x | G ≥ . • √ x G G = | x | G • | e y | G = e | y | • | x ⊙ y | G = | x | G ⊙ | y | G • | x ⊕ y | G ≤ | x | G ⊕ | y | G • | x ⊘ y | G = | x | G ⊘ | y | G • | x ⊖ y | G ≥ | x | G ⊖ | y | G • G ⊖ G ⊙ ( x ⊖ y ) = y ⊖ x i.e. in short ⊖ ( x ⊖ y ) = y ⊖ x. Further e − x = ⊖ e x holds for all x ∈ Z + . Thus the set of all geometric integers turns outto the following: Z ( G ) = { ...., e − , e − , e − , e , e , e , e , .... } = { ...., ⊖ e , ⊖ e , ⊖ e, , e, e , e , .... } . Definitions and Notations
Now we recall some definitions and results discussed in [4, 5].3.1.
Geometric Binomial Formula. ( i ) ( a ⊕ b ) G = a G ⊕ e ⊙ a ⊙ b ⊕ b G . ( ii ) ( a ⊕ b ) G = a G ⊕ e ⊙ a G ⊙ b ⊕ e ⊙ a ⊙ b G ⊕ b G . In general ( iii ) ( a ⊕ b ) n G = a n G ⊕ e ( n ) ⊙ a ( n − G ⊙ b ⊕ e ( n ) ⊙ a ( n − G ⊙ b G ⊕ .... ⊕ b n G = G n X r =0 e ( nr ) ⊙ a ( n − r ) G ⊙ b r G . Similarly ( a ⊖ b ) n G = G n X r =0 ( ⊖ e ) r G ⊙ e ( nr ) ⊙ a ( n − r ) G ⊙ b r G . Boruah and Hazarika e -4 e -3 e -2 e -1 ee e e e-4 e -3 e -2 e -1 e e Y' G Y G X G X' G Figure 1.
Geometric Co-ordinate System
Note : x ⊕ x = x . Also e ⊙ x = x ln( e ) = x . So, e ⊙ x = x = x ⊕ x. Geometric Real Number Line.
For x, y ∈ R ( G ) , there exist u, v ∈ R such that x = e u and y = e v . Also consecutive natural numbers are equally spaced by one unit in realnumber line, but the geometric integers e, e , e , ... are not equally spaced in ordinary sense,e.g. e − e = 4 . e − e = 12 . e ⊖ e = e − = e, e ⊖ e = e − = e etc. Furthermore, it can be easilyverified that ( R ( G ) , ⊕ , ⊙ ) is a complete field with geometric identity e and geometric zero1 . So we can consider a new type of geometric real number line.3.3.
Geometric Co-ordinate System.
We consider two mutually perpendicular geo-metric real number lines which intersect each other at (1 ,
1) as shown in FIGURE 1.Since consecutive geometric integers are equidistant in geometric sense and ( R ( G ) , ⊕ , ⊙ )is a complete field, so almost all the properties of ordinary cartesian coordinate systemwill be valid for geometric coordinate system under geometric arithmetic.3.4. Geometric Factorial.
In [4], we defined geometric factorial notation ! G as n ! G = e n ⊙ e n − ⊙ e n − ⊙ · · · ⊙ e ⊙ e = e n ! . Generalized Geometric Forward Difference Operator.
Let∆ G f ( a ) = f ( a ⊕ h ) ⊖ f ( a ) . ∆ G f ( a ) = ∆ G f ( a ⊕ h ) ⊖ ∆ G f ( a )= f ( a ⊕ e ⊙ h ) ⊖ e ⊙ f ( a ⊕ h ) ⊕ f ( a ) . ∆ G f ( a ) = ∆ G f ( a ⊕ h ) ⊖ ∆ G f ( a )= f ( a ⊕ e ⊙ h ) ⊖ e ⊙ f ( a ⊕ e ⊙ h ) ⊕ e ⊙ f ( a ⊕ h ) ⊖ f ( a ) . Thus, n th forward difference is∆ nG f ( a ) = G n X k =0 ( ⊖ e ) k G ⊙ e ( nk ) ⊙ f ( a ⊕ e n − k ⊙ h ) , with ( ⊖ e ) G = e -Calculus 5 Generalized Geometric Backward Difference Operator.
Let ∇ G f ( a ) = f ( a ) ⊖ f ( a ⊖ h ) . ∇ G f ( a ) = ∇ G f ( a ) ⊖ ∇ G f ( a ⊖ h )= f ( a ) ⊖ e ⊙ f ( a ⊖ h ) ⊕ f ( a ⊖ e ⊙ h ) . ∇ G f ( a ) = ∇ G f ( a ) ⊖ ∇ G f ( a − h )= f ( a ) ⊖ e ⊙ f ( a ⊖ h ) ⊕ e ⊙ f ( a ⊖ e ⊙ h ) ⊖ f ( a ⊖ e ⊙ h ) . Thus, n th geometric backward difference is ∇ nG f ( a ) = G n X k =0 ( ⊖ e ) k G ⊙ e ( nk ) ⊙ f ( a ⊖ e k ⊙ h ) . New Results
Geometric Pythagorean Triplets.
Three numbers x, y, z ∈ R ( G ) are said to beformed a geometric Pythagorean triplet if x G = y G ⊕ z G . (4.1)Or, equivalently x ln x = y ln y .z ln z . Taking natural log to both sides, we get(ln x ) = (ln y ) + (ln z ) . Thus, if { x, y, z } ⊂ R ( G ) is a geometric Pythagorean triplet(GPT), then { ln x, ln y, ln z } forms an ordinary Pythagorean triplet(OPT). Conversely, if { a, b, c } is an OPT, then { e a , e b , e c } forms a GPT where a, b, c ∈ R are non-negative. For example, { , , } is anOPT as 5 = 3 + 4 . So, { e , e , e } is a GPT. Since, for any given positive integer m wecan form an OPT as { m − , m, m + 1 } , similarly we can form infinite number of GPT. Definition 4.1.1 (Geometric Right Triangle) . In the geometric co-ordinate system, ifgeometric lengths of the three sides of a triangle represent a GPT, then the triangle willbe called geometric right triangle. i.e. if h, p, b are lengths of the three sides of a trianglesuch that h G = p G ⊕ b G , then the triangle is called a geometric right triangle withhypotenuse h. It is to be noted that a GPT does not form a triangle with respect to the ordinaryco-ordinate system. For example, GPT { e , e , e } never forms a triangle as e + e < e , i.e. sum of two sides is less than the third side, which is impossible for a triangle. Also,to form a geometric triangle, length of each side must be greater than 1 . Definition 4.1.2.
Area of geometric right triangle = ln (cid:16) √ base ⊙ altitude (cid:17) = ln(base) . ln(altitude)2 . Boruah and Hazarika
Now we define a new trigonometric ratios with respect to the geometric right triangle.This geometric trigonometry will give us the geometrical interpretation of G-calculus.With respect to geometric right triangle, denote trigonometric ratios sin , cos , tan , cot , secand csc respectively as sing , cosg , tang , cotg , secg and cscg . Geometric Trigonometric Ratios.
Let θ be an acute angle of a geometric righttriangle and length of the sides be h, p, b ∈ R ( G ) , respectively such that h = hypotenuse p = side opposite to the angle θb = side adjacent to the angle θ. Then we define sing θ = ph G = p h cscg θ = hp G = h p cosg θ = bh G = b h secg θ = hb G = h b tang θ = pb G = p b cotg θ = bp G = b p Figure 2
Relation between geometric trigonome-try and ordinary trigonometry.
It is clear that n unit length in ordinary coordinate system is equalto e n unit in geometric coordinate system. So prop-erties of the ordinary right triangle having sides oflengths 3 , , e , e , e respectively as shown in the FIG-URE 2. That is, area of the both the triangles = 6square unit, ∠ A = 36 . ◦ , ∠ B = 53 . ◦ and ∠ C = 90 ◦ . Similarly, properties of the geo-metric right triangle having sides h, p, b ∈ R ( G ) will be same to the ordinary right trianglehaving sides h ′ = ln( h ) , p ′ = ln( p ) and b ′ = ln( b ) , respectively. Nowsing θ = p h ) = e ln (cid:20) p h ) (cid:21) = e ln( p )ln( h ) = e p ′ h ′ = e sin θ . Similarly, cosg θ = e cos θ , tang θ = e tan θ etc. Also,sing θ cosg θ G = ph G ⊙ hb G= pb G= tang θ. -Calculus 7 Geometric Trigonometric Identities.
We can easily verify that all the identitiesof ordinary trigonometry for acute angle are also valid for geometric trigonometry withrespect to the geometric arithmetic system ( ⊕ , ⊖ , ⊙ , ⊘ ) . For,sing G A ⊕ cosg G A = e sin A ⊙ e sin A ⊕ e cos A ⊙ e cos A = e sin A ⊕ e cos A = e sin A .e cos A = e ( sin A +cos A ) = e = e. Thus, sing A ⊙ cscg A = e, sing G A ⊕ cosg G A = e cos A ⊙ sec A = e, tang G A ⊕ e = secg G A tang A ⊙ cotg A = e, cotg G A ⊕ e = cscg G A. Again sing( A + B ) = e sin( A + B ) = e (sin A cos B +cos A sin B ) = e sin A cos B .e cos A sin B = e sin A cos B ⊕ e cos A sin B = e sin A ⊙ e cos B ⊕ e cos A ⊙ e sin B = sing A ⊙ cosg B ⊕ cosg A ⊙ sing B. Similarly, cosg( A + B ) = cosg A ⊙ cosg B ⊖ sing A ⊙ sing B. Here, a question may arise that why the ordinary sum of the angles A + B is taken insteadof the geometric sum A ⊕ B. Cause is that, measure of the angles is invariant under theboth ordinary and geometric coordinate systems. Of course, for practical uses, geometricsum will be essential, specially for G -derivative of trigonometric functions.4.5. G -Limit. According to Grossman and Katz[9], geometric limit of a positive valuedfunction defined in a positive interval is same to the ordinary limit. Here, we define G -limit of a function with the help of geometric arithmetic as follows:A function f, which is positive in a given positive interval, is said to tend to the limit l > x tends to a ∈ R , if, corresponding to any arbitrarily chosen number ǫ > , however small(but greater than 1), there exists a positive number δ > , such that1 < | f ( x ) ⊖ l | G < ǫ for all values of x for which 1 < | x ⊖ a | G < δ. We write G lim x → a f ( x ) = l or f ( x ) G −→ l. Boruah and Hazarika
Here, | x ⊖ a | G < δ ⇒ (cid:12)(cid:12)(cid:12) xa (cid:12)(cid:12)(cid:12) G < δ ⇒ δ < xa < δ ⇒ aδ < x < aδ. Similarly, | f ( x ) ⊖ l | G < ǫ ⇒ lǫ < f ( x ) < lǫ. Thus, in ordinary sense, f ( x ) G −→ l means that for any given positive real number ǫ > , no matter however closer to 1 , ∃ a finite number δ > f ( x ) ∈ ] lǫ , lǫ [ for every x ∈ ] aδ , aδ [ . It is to be noted that lengths of the open intervals ] aδ , aδ [ and ] lǫ , lǫ [ decreasesas δ and ǫ respectively decreases to 1 . Therefore, as ǫ decreases to 1 , f ( x ) becomes closerand closer to l, as well as x becomes closer and closer to a as δ decreases to 1 . Hence, l is also the ordinary limit of f ( x ) . i.e. f ( x ) G −→ l ⇒ f ( x ) → l. In other words, we cansay that G -limit and ordinary limit are same for bipositive functions whose functionalvalues as well as arguments are positive in the given interval. Only difference is that in G -calculus we approach the limit geometrically, but in ordinary calculus we approach thelimit linearly.A function f is said to tend to limit l as x tends to a from the left, if for each ǫ > δ > | f ( x ) ⊖ l | G < ǫ when a/δ < x < a. Insymbols, we then write G lim x → a − f ( x ) = l or f ( a −
1) = l. Similarly, a function f is said to tend to limit l as x tends to a from the right, if for each ǫ > δ > | f ( x ) ⊖ l | G < ǫ when a < x < aδ. Insymbols, we then write G lim x → a + f ( x ) = l or f ( a + 1) = l. If f ( x ) is negative valued in a given interval, it will be said to tend to a limit l < ǫ > , ∃ δ > f ( x ) ∈ ] lǫ, lǫ [ whenever x ∈ ] aδ , aδ [ . G -Continuity. A function f is said to be G -continuous at x = a if(i) f ( a ) i.e., the value of f ( x ) at x = a, is a definite number,(ii) the G -limit of the function f ( x ) as x G −→ a exists and is equal to f ( a ) . Alternatively, a function f is said to be G -continuous at x = a, if for arbitrarily chosen ǫ > , however small, there exists a number δ > | f ( x ) ⊖ f ( a ) | G < ǫ for all values of x for which, | x ⊖ a | G < δ. On comparing the above definitions of limits and continuity, we can conclude that afunction f is G -continuous at x = a iflim x → a f ( x ) f ( a ) = 1 . -Calculus 9 Basic Properties of G -Calculus G -Derivative and its Interpretation. In [5] we defined the G -differentiation of f ( x ) as d G fdx = f G ( x ) = G lim h → f ( x ⊕ h ) ⊖ f ( x ) h G for h ∈ R ( G ) . (5.1)Equivalently d G fdx = G lim h → f ( x ⊕ h ) ⊖ f ( x ) h G = lim h → (cid:20) f ( hx ) f ( x ) (cid:21) h = lim u → (cid:20) f ( e u .x ) f ( x ) (cid:21) u where h = e u ∈ R ( G )Here, what ever we deduce, can be expressible in terms of geometric arithmetical system,though we express results in terms of classical arithmetic for easy comparison. So, the G -derivative of a positive valued function f at a point c belonging to a positive intervalcan be defined as f G ( c ) = G lim x → c f ( x ) ⊖ f ( c ) x ⊖ c G or f G ( c ) = lim x → c (cid:20) f ( x ) f ( c ) (cid:21) xc ) . (5.2)Equation (5.2) is the bigeometric slope define by Grossman in [10]. Instead of the phase“bigeometric calculus” term “ G -calculus” is used because, depending on Grossman [10]and Grossman and Katz’s [11] pioneering works, we are trying to develop his work withthe help of geometric arithmetic system.From (5.2), it is clear that G -derivative exist if both f ( x ) and f ( c ) takes same sign andat the same time x and c takes same sign.We know that x + h is arithmetic change to x. Here, x ⊕ h is geometric change to theindependent variable x. We are saying x ⊕ h is geometric change because x, x ⊕ h, x ⊕ h , h ⊕ h , .... forms a geometric progression x, xh, xh , xh , .... just as x, x + h, x + 2 h, x + 3 h, .... forms an arithmetic progression. Now, as the independent variable changes from x to xh (i.e. to x ⊕ h ), value of the function changes from f ( x ) to f ( x ⊕ h ) = f ( xh ) . Geometricchange to x is given by ∆ x = x ⊕ h ⊖ x = xhh = h whereas geometric change to y = f ( x ) is given by∆ y = f ( x ⊕ h ) ⊖ f ( x ) = f ( xh ) f ( x ) In case of ordinary derivative ∆ y ∆ x = f ( x + h ) − f ( x ) h gives the average additive change in f ( x )per unit change in x over the interval [ x, x + ∆ x ] = [ x, x + h ] . Here in G -calculus,∆ y ∆ x G = (∆ y ) x ) = (cid:20) f ( xh ) f ( x ) (cid:21) h gives the average geometric change in f ( x ) per unit geometric change in x over the interval[ x, xh ] . Now, if we take the limit as ∆ x (i.e. h ) tends to the geometric zero,1 , we get d G yd G x = G lim ∆ x → ∆ y ∆ x G = G lim ∆ x → (∆ y ) x ) = lim h → (cid:20) f ( xh ) f ( x ) (cid:21) h . Or, in short f G ( x ) = ( d G y ) dGx ) = lim h → (cid:20) f ( xh ) f ( x ) (cid:21) h . It is to be noted that G-derivative exists if f ( x ) = 0 and f ( x ) , f ( hx ) are both positive orboth negative. Also, it is obvious that d G yd G x = dydx G = dy dx ) . It is obvious that y = m ⊙ x ⊕ c i.e. y = c.x ln m represents a straight line with slope m ingeometric co-ordinate system as well as in log-log paper. Then, dy G dx G = m. i.e. G-derivativeis the slope of the geometric straight line. Note:
For the convenience, we use the symbol f [ n ] to denote n th geometric derivative f ( n G ) . For example, the second geometric derivative of f ( x ) is given by d G fdx G = f [2] ( x ) = G lim h → f G ( x ⊕ h ) ⊖ f G ( x ) h G. Alternatively, G -derivative at point x = c can be written as f G ( x ) = lim x → c (cid:20) f ( x ) f ( c ) (cid:21) xc ) . We call that left hand G -derivative and right hand G -derivative exist at x = c iflim x → c − (cid:18) f ( c.h ) f ( c ) (cid:19) xc ) and lim x → c + (cid:18) f ( c.h ) f ( c ) (cid:19) xc ) . exist, respectively. Theorem 5.1.1.
If a function f is G -differentiable and is positive, then it is both G -continuous and ordinarily continuous.Proof. Let f G ( x ) exists, where f G ( x ) = lim x → c (cid:20) f ( x ) f ( c ) (cid:21) xc ) . -Calculus 11 Now, lim x → c f ( x ) f ( c ) = lim x → c "(cid:18) f ( x ) f ( c ) (cid:19) xc ) ln( xc ) = (cid:2) f G ( c ) (cid:3) ln( xc ) = (cid:2) f G ( c ) (cid:3) = 1 . Hence, G -derivative ⇒ G -Continuous. Now, for f ( c ) = 0 , we havelim x → c f ( x ) f ( c ) = 1 ⇒ lim x → c f ( x ) f ( c ) − ⇒ lim x → c f ( x ) − f ( c ) f ( c ) = 0 ⇒ lim x → c f ( x ) − f ( c ) = 0 , since f ( c ) = 0 ⇒ lim x → c f ( x ) = f ( c ) . Thus, if G -limit of f ( x ) exists at x = c, then its ordinary limit exists and is equal to f ( c ) . (cid:3) Proposition 5.1.2.
A continuous function f is not necessarily G -derivable.Proof. Let us consider the function f ( x ) = | x | G = x, if x > , if x = 1 x , if 0 < x < . Then, obviously it is continuous at x = 1 . But, we show that it is not G -differentiable at x = 1 . Left hand G -derivative is given bylim h → (cid:18) f (1 .h ) f (1) (cid:19) h = lim h → | h | G ! h = lim h → h h = e. Right hand G -derivative is given bylim h → − (cid:18) f (1 .h ) f (1) (cid:19) h = lim h → − | h | G ! h = lim h → (cid:18) h (cid:19) h = 1 e . Thus Lf G (1) = Rf G (1) . (cid:3) Example . If f ( x ) = x n G , then f G ( x ) = e n ⊙ x ( n − G and f ( n G ) = e n ! . Example . Let f ( x ) = x. Then d G fdx G = e. Example . If f ( x ) = e x , then f G ( x ) = e x . Proof. f G ( x ) = G lim h → e x ⊕ h ⊖ e x h G = lim h → e xh − x h G = lim h → (cid:2) e xh − x (cid:3) h = lim h → e x ( h − h = e x , since by ordinary L’ Hospital rule, lim h → ( h − h = 1 . (cid:3) Example . G -derivative of f ( x ) = x n is constant, where n is a positive integer. Proof. f G ( x ) = G lim h → ( x ⊕ h ) n ⊖ x n h G = lim h → (cid:20) x n h n x n (cid:21) h = lim h → (cid:16) h h (cid:17) n = e n , since h h = e = a constant . (cid:3) Example . If f ( x ) = sin x, then f G ( x ) = e x cot x . Proof. f G ( x ) = G lim h → sin( x ⊕ h ) ⊖ sin xh G = lim h → (cid:20) sin( xh )sin x (cid:21) h (1 ∞ form)= lim h → e ln [ sin( xh )sin x ] h = e lim h → [ ln(sin( xh )) − ln(sin x )ln h ] ( 00 form)= e lim h → [ h.x. cos( xh )sin( xh ) ] (differentiating numerator and denominator w.r.t. h )= e x cot x . -Calculus 13 (cid:3) Remark . Though f ( x ) = x n is a polynomial of degree n in ordinary sense, butgeometrically it is a polynomial of degree one as x n = e n ⊙ x. So, its G -derivative isconstant for any positive integer n. Relation between G -derivative and ordinary derivative. By definition, G -derivative of a positive valued function f ( x ) is given by f G ( x ) = G lim h → f ( x ⊕ h ) ⊖ f ( x ) h G = lim h → (cid:20) f ( hx ) f ( x ) (cid:21) h , which is in 1 ∞ indeterminate form . Using logarithm, to transform it to indeterminate form and then applying L’ Hospitalrule, we can make a relation between G -derivative and ordinary derivative as follows: f G ( x ) = lim h → e ln [ f ( hx ) f ( x ) ] h = lim h → e ln f ( hx ) − ln f ( x )ln h = e lim h → [ ln f ( hx ) − ln f ( x )ln h ] , (since the exponential function is continuous)= e lim h → (cid:20) ddh f ( hx ) f ( hx ) / h (cid:21) , (applying L’ Hospital rule)= e lim h → hxf ′ ( hx ) f ( hx ) , (since ddh f ( hx ) = ddx f ( hx ))= e xf ′ ( x ) f ( x ) . Thus, f G ( x ) = e x f ′ ( x ) f ( x ) . (5.3)Instead of using the definition of G-derivative, often we’ll use the relation (5.3).5.3. G -derivatives of some standard functions. • G -derivative of a constant: If f ( x ) = c, then d G dx G ( f ( x )) = e x f ′ ( x ) f ( x ) = e = 1 • G -derivative of ordinary product of a constant and a function: d G dx G ( cf ( x )) = e x cf ′ ( x ) cf ( x ) = e x f ′ ( x ) f ( x ) = d G dx G ( f ( x )) . • G -derivative of ordinary product of two functions: d G dx G ( f ( x ) .g ( x )) = e x f ( x ) .g ′ ( x )+ f ′ ( x ) .g ( x ) f ( x ) .g ( x ) = e x f ′ ( x ) f ( x ) .e x g ′ ( x ) g ( x ) = d G dx G ( f ( x )) . d G dx G ( g ( x )) . or, d G dx G ( f ( x ) ⊕ g ( x )) = d G dx G ( f ( x )) ⊕ d G dx G ( g ( x )) . (5.4) • G -derivative of quotient of two functions: d G dx G (cid:18) f ( x ) g ( x ) (cid:19) = e x. g ( x ) f ( x ) . g ( x ) .f ′ ( x ) − f ( x ) .g ′ ( x ) g x ) = e x. g ( x ) .f ′ ( x ) − f ( x ) .g ′ ( x ) f ( x ) .g ( x ) = e x f ′ ( x ) f ( x ) e x g ′ ( x ) g ( x ) = d G dx G ( f ( x )) d G dx G ( g ( x )) . or, d G dx G ( f ( x ) ⊖ g ( x )) = d G dx G ( f ( x )) ⊖ d G dx G ( g ( x )) . (5.5) • G -derivative of trigonometric functions: d G dx G (sin x ) = e x cot x , d G dx G (cot x ) = e − x sec x csc x d G dx G (cos x ) = e − x tan x , d G dx G (sec x ) = e x tan x d G dx G (tan x ) = e x sec x csc x , d G dx G (csc x ) = e − x cot x . • G -derivative of sum and product functions: . ( u ⊙ v ) G = u G ⊙ v ⊕ u ⊙ v G . In ordinary sense, (cid:0) u ln v (cid:1) G = (cid:0) u G (cid:1) ln v . (cid:0) v G (cid:1) ln u . . ( e u ) G = e xf ′ . . ( f + g ) G = e x ( f ′ + g ′ ) f + g = (cid:0) f G (cid:1) ff + g . (cid:0) g G (cid:1) gf + g . . dy G dx G ( f ◦ g ) ( x ) = e x.f ′ [ g ( x )] .g ′ ( x ) f [ g ( x )] . Remark . The function f ( x ) = e x remains unchanged under both ordinary deriva-tive and G -derivative. It is observed that ordinary derivatives of the ordinary Taylor’sexpansion of f ( x ) = e x is invariant as f ( x ) = e x = 1 + x + x
2! + x
3! + x
4! + ... = f ′ ( x ) = f ′′ ( x ) = ... Similar to ordinary derivative, in [5] we have proved that the n th G -derivative of a geo-metric polynomial of degree n is constant. Since e x remains unchanged under any numberof G -derivative, it must have infinite geometric polynomial expansion which will remainunchanged under geometric differentiations. Next, we try to give geometric Taylor’s ex-pansions of different functions. Theorem 5.3.1. If f : ( a, b ) : −→ R ( G ) is G -differentiable, then (i) f is increasing, if f G ≥ . (ii) f is decreasing, if f G ≤ . -Calculus 15 Proof.
Let c be an interior point of the domain [ a, b ] of a function f and f G ( c ) exists andbe positive, i.e. f G ( c ) > . By definition of G -derivative,lim x → c (cid:20) f ( x ) f ( c ) (cid:21) x/c ) = f G ( c ) , x = c. i.e. f G ( c ) is the limit of h f ( x ) f ( c ) i x/c ) . Then for given ǫ > , ∃ δ > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) f ( x ) f ( c ) (cid:21) x/c ) ⊖ f G ( c ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G < ǫ, where | x ⊖ c | G < δ, x = c ⇒ f G ( c ) ǫ < (cid:20) f ( x ) f ( c ) (cid:21) x/c ) < ǫ.f G ( c ) . If ǫ > ǫ < f G ( c ) , then h f ( x ) f ( c ) i x/c ) > f G ( c ) ǫ > , where x ∈ ] c/δ, cδ [ . Then(i) f ( x ) f ( c ) > , i.e. f ( x ) > f ( c ) if x ∈ ] c, cδ [,(ii) f ( x ) f ( c ) < , i.e. f ( x ) < f ( c ) if x ∈ ] c/δ, c [.Thus from (i) and (ii) f ( x ) is increasing at x = c. Hence the function is increasing at x = c if f G ( c ) > . Similarly, it can be proved that the function is decreasing at x = c if f G ( c ) < . (cid:3) Theorem 5.3.2 (Darboux’s Theorem) . If a function f is G -derivable on a closed interval [ a, b ] and f G ( a ) , f G ( b ) are of opposite signs (i.e. one is > , other is < ) then there existsat least one point c between a and b such that f G ( c ) = 0 . Proof.
Let f G ( a ) < f G ( b ) > . Since, G -derivative exists ⇒ ordinary derivativeexists, so, f ′ ( a ) and f ′ ( b ) exist. Now f G ( a ) < ⇒ e a f ′ ( a ) f ( a ) < ⇒ a f ′ ( a ) f ( a ) < ⇒ f ′ ( a ) < . Similarly, f G ( b ) > ⇒ f ′ ( b ) > . Therefore from Newtonian calculus, there exists c ∈ [ a, b ] s.t. f ′ ( c ) = 0 and so f G ( c ) = e c f ′ ( c ) f ( c ) = 1 . (cid:3) Theorem 5.3.3 (Intermediate value theorem for derivatives) . If a function f is G -derivable on a closed interval [ a, b ] and f G ( a ) = f G ( b ) and k be a number lying btween f G ( a ) and f G ( b ) , then ∃ at least one point c ∈ ] a, b [ such that f G ( c ) = k. Proof.
Let g ( x ) = f ( x ) x ln k . Then by the rule of G -derivative of quotient of two functions g G ( a ) = f G ( a ) k and g G ( b ) = f G ( b ) k . Since f G ( a ) < k < f G ( b ) , so f G ( a ) k and f G ( b ) k can not be greater than 1 at the same time.Therefore, if g G ( a ) > g G ( b ) < . Hence, g ( x ) satisfies the conditions of Darboux’stheorem. Thus, there exists at least one point c ∈ ] a, b [ such that g G ( c ) = 1 , i.e. f G ( c ) = k. (cid:3) Theorem 5.3.4 (Rolle’s Theorem) . If a function f defined on [ a, b ] is (i) G -continuous on [ a, b ] , (ii) G -derivable on ] a, b [ , (iii) f ( a ) = f ( b ) , then there exists at least one number c between a and b such that f G ( c ) = 1 . Proof.
Since G -continuous functions are ordinary continuous and f ′ ( x ) exists if f G ( x )exists. So, f satisfies the conditions of Rulle’s theorem of Newtonian calculus. So, thereexists c ∈ ] a, b [ such that f ′ ( c ) = 0 . Hence f G ( c ) = e cf ′ ( c ) f ( c ) = 1 . (cid:3) Theorem 5.3.5 (Lagrange’s Mean Value Theorem) . If a function f defined on [ a, b ] is (i) G -continuous on [ a, b ] , (ii) G -derivable on ] a, b [ , then there exists at least one c ∈ ] a, b [ such that f G ( c ) = (cid:20) f ( b ) f ( a ) (cid:21) ba ) Proof.
Les us define a function φ ( x ) = x ln k .f ( x )where the constant k is so determined that φ ( a ) = φ ( b ) .φ ( a ) = φ ( b ) ⇒ a ln k .f ( a ) = b ln k .f ( b ) ⇒ h ab i ln k = f ( b ) f ( a ) . Using natural logarithm to both sides we get k = (cid:20) f ( b ) f ( a ) (cid:21) ab ) = (cid:20) f ( b ) f ( a ) (cid:21) − ba ) . Now, the function φ ( x ) , the product of two G -derivable and G -continuous functions, isitself(i) G -continuous on [ a, b ] , (ii) G -derivable on ] a, b [ , and(iii) φ ( a ) = φ ( b ) . -Calculus 17 Therefore by Rolle’s theorem ∃ c ∈ ] a, b [ such that φ G ( c ) = 1 . But φ G ( x ) = d G dx G ( x ln k ) . d G dx G ( f ( x )) , by the rule of G -derivative of product function, φ G ( x ) = k.f G ( x ) . ⇒ φ G ( c ) = k.f G ( c ) f G ( c ) = 1 k = (cid:20) f ( b ) f ( a ) (cid:21) ba ) . In geometric sense f G ( c ) = f ( b ) ⊖ f ( a ) b ⊖ a G. (cid:3) Note:
In the above theorem, if we replace b by ah, where h > , then the number c between a and b may be taken as a.h ln θ for 1 < θ < e. Thus f G ( a.h ln θ ) = (cid:20) f ( ah ) f ( a ) (cid:21) aha ) or f ( ah ) = f ( a ) . (cid:2) f G ( a.h ln θ ) (cid:3) ln h , where 1 < θ < e. or f ( a ⊕ h ) = f ( a ) ⊕ h ⊙ f G ( a ⊕ h.θ ) . Now we deduce geometric Taylor’s expansion for f ( ah ) with the help of Rolle’s Theorem.Firstly, we have to find G -derivative of two important functions as follows. Lemma 5.3.6. If y = (cid:2) f [ n ] ( x ) (cid:3) ln n ( ahx ) n ! then y G = d G dx G [ y ] = [ f [ n +1] ( x ) ] ln n ( ahx ) n ! [ f [ n ] ( x ) ] ln( n − ahx )( n − Proof. y = (cid:2) f [ n ] ( x ) (cid:3) ln n ( ahx ) n ! Taking logarithm to both sides, we getln y = ln f [ n ] ( x ) . ln n ( ahx ) n ! ⇒ y ′ y = ddx (cid:0) f [ n ] ( x ) (cid:1) f [ n ] ( x ) . ln n ( ahx ) n ! + ln f [ n ] ( x ) . ln n − ( ahx )( n − . − ahx ahx (differentiating w.r.t. x ) ⇒ e x y ′ y = e x f ′ [ n ]( x ) f [ n ]( x ) . ln n ( ahx ) n ! .e x ln f [ n ] ( x ) . ln( n − ahx )( n − . − x ⇒ y G = " e x f ′ [ n ]( x ) f [ n ]( x ) ln n ( ahx ) n ! .e ln [ f [ n ] ( x ) ] − ln( n − ahx )( n − ⇒ y G = (cid:2) f [ n +1] ( x ) (cid:3) ln n ( ahx ) n ! . (cid:2) f [ n ] ( x ) (cid:3) − ln( n − ahx )( n − ⇒ y G = (cid:2) f [ n +1] ( x ) (cid:3) ln n ( ahx ) n ! [ f [ n ] ( x )] ln( n − ahx )( n − . (cid:3) Lemma 5.3.7. If y = k ln p ( ahx ) where k is a constant and p is a positive integer, then y G = k − p ln ( p − ( ahx ) . Proof.
Taking logarithm to the both sides of y = k ln p ( ahx ) , we getln y = ln p ( ahx ) . ln k ⇒ y ′ y = ln k.p ln ( p − ( ahx ) . − ahx ahx (differentiating w.r.t. x ) ⇒ e x y ′ y = k − p ln ( p − ( ahx ) ⇒ y G = k − p ln ( p − ( ahx ) . (cid:3) Theorem 5.3.8 (Geometric Taylor’s Theorem) . If a function defined on [ a, ah ] is suchthat (i) the ( n − th G -derivative of f, i.e. f [ n − is G -continuous on [ a, ah ] , and (ii) the n th G -derivative, f [ n ] exists on [ a, ah ] then there exists at least one number θ between and e such that f ( ah ) = f ( a ) . (cid:2) f [1] ( a ) (cid:3) ln h . (cid:2) f [2] ( a ) (cid:3) ln2 h . (cid:2) f [3] ( a ) (cid:3) ln3 h ...... (cid:2) f [ n − ( a ) (cid:3) ln n − h ( n − . (cid:2) f [ n ] ( a.h ln θ ) (cid:3) (1 − ln θ )( n − p ) ln n h ( n − p (5.6) -Calculus 19 Proof.
Condition ( i ) in the statement implies that f [1] , f [2] , f [3] , ..., f [ n − exists and arecontinuous on [ a, ah ] . Let us consider the function φ ( x ) = f ( x ) . (cid:2) f [1] ( x ) (cid:3) ln( ahx ) . (cid:2) f [2] ( x ) (cid:3) ln2( ahx )2! . (cid:2) f [3] ( x ) (cid:3) ln3( ahx )3! ...... (cid:2) f [ n − ( x ) (cid:3) ln n − ahx )( n − .A ln p ( ahx ) (5.7)where A is a constant to be determined such that φ ( ah ) = φ ( a ) . But, putting x = ah and x = a in (5.7), respectively, we get φ ( ah ) = f ( ah ) , and φ ( a ) = f ( a ) . (cid:2) f [1] ( a ) (cid:3) ln h . (cid:2) f [2] ( a ) (cid:3) ln2 h ... (cid:2) f [ n − ( a ) (cid:3) ln n − h ( n − .A ln p h . ∴ f ( ah ) = f ( a ) . (cid:2) f [1] ( a ) (cid:3) ln h . (cid:2) f [2] ( a ) (cid:3) ln2 h ... (cid:2) f [ n − ( a ) (cid:3) ln n − h ( n − .A ln p h . (5.8)Now(i) f, f [1] , f [2] , f [3] , ..., f [ n − all being continuous on [ a, ah ], the function φ ( x ) is con-tinuous on [ a, ah ];(ii) the functions f, f [1] , f [2] , f [3] , ..., f [ n − and ln r ( ahx ) for all r being derivable in ] a, ah [ , the function φ ( x ) is derivable in ] a, ah [;(iii) φ ( ah ) = φ ( a ) . Hence, φ ( x ) satisfies all the conditions of Rolle’s Theorem and hence there exists one realnumber θ ∈ ]1 , e [ such that φ G ( a.h ln θ ) = 1 . Now, using Lemma 5.3.6 and Lemma 5.3.7 φ G ( x ) = f [1] ( x ) . (cid:2) f [2] ( x ) (cid:3) ln( ahx ) f [1] ( x ) . (cid:2) f [3] ( x ) (cid:3) ln2( ahx )2! [ f [2] ( x )] ln( ahx ) . (cid:2) f [4] ( x ) (cid:3) ln3( ahx )3! [ f [3] ( x )] ln2( ahx )2! ...... (cid:2) f [ n ] ( x ) (cid:3) ln( n − ahx )( n − [ f [ n − ( x )] ln( n − ahx )( n − .A − p ln ( p − ( ahx ) which gives φ G ( x ) = (cid:2) f [ n ] ( x ) (cid:3) ln( n − ahx )( n − .A − p ln ( p − ( ahx ) ⇒ φ G ( a.h ln θ ) = (cid:2) f [ n ] ( a.h ln θ ) (cid:3) ln( n − aha.h ln θ )( n − .A − p ln( p − aha.h ln θ ) ⇒ A p [ ln( h − ln θ ) ] ( p − = (cid:2) f [ n ] ( a.h ln θ ) (cid:3) [ ln( h − ln θ ) ] ( n − n − ⇒ A p [(1 − ln θ ) ln h ] ( p − = (cid:2) f [ n ] ( a.h ln θ ) (cid:3) [(1 − ln θ ) ln h ]( n − n − ⇒ A = (cid:2) f [ n ] ( a.h ln θ ) (cid:3) [(1 − ln θ ) ln h ]( n − p [(1 − ln θ ) ln h ]( p − n − ⇒ A = (cid:2) f [ n ] ( a.h ln θ ) (cid:3) [(1 − ln θ ) ln h ]( n − p )( n − p ⇒ A = (cid:2) f [ n ] ( a.h ln θ ) (cid:3) (1 − ln θ )( n − p ) ln( n − p ) h ( n − p . Now substituting the value of A in (5.8), we get f ( ah ) = f ( a ) . (cid:2) f [1] ( a ) (cid:3) ln h . (cid:2) f [2] ( a ) (cid:3) ln2 h ... (cid:2) f [ n − ( a ) (cid:3) ln n − h ( n − . (cid:2) f [ n ] ( a.h ln θ ) (cid:3) (1 − ln θ )( n − p ) ln n h ( n − p . (5.9) (cid:3) Geometric Taylor’s Series.
In (5.9), the term R n = (cid:2) f [ n ] ( a.h ln θ ) (cid:3) (1 − ln θ )( n − p ) ln n h ( n − p is called Taylor’s remainder after n terms. Since, 0 < − ln θ < < θ < e, so,(1 − ln θ ) n − p → n → ∞ . Therefore, if f possesses G -derivative of every order in[ a, ah ] then R n → n → ∞ . Then Taylor’s expansion becomes f ( ah ) = f ( a ) . (cid:2) f [1] ( a ) (cid:3) ln h . (cid:2) f [2] ( a ) (cid:3) ln2 h ... (cid:2) f [ n ] ( a ) (cid:3) ln n hn ! ... = Π ∞ n =0 (cid:2) f [ n ] ( a ) (cid:3) ln n hn ! . (5.10)This expression can be written in terms of geometric operations as f ( a ⊕ h ) = f ( a ) ⊕ h ⊙ f [1] ( a ) ⊕ h G G G ⊙ f [2] ( a ) ⊕ ... ⊕ h n G n ! G G ⊙ f [ n ] ( a ) ⊕ ... = G ∞ X n =0 h n G n ! G G ⊙ f [ n ] ( a ) , (5.11)where n ! G = e n ! and h n G = h ln ( n − h . The equivalent expressions (5.10) and (5.11) will becalled respectively as Taylor’s product and Geometric Taylor’s series. If we put a = 1 and h = x in (5.10), we get f ( x ) = f (1) . (cid:2) f [1] (1) (cid:3) ln x . (cid:2) f [2] (1) (cid:3) ln2 x ... (cid:2) f [ n ] (1) (cid:3) ln n xn ! ... = Π ∞ n =0 (cid:2) f [ n ] (1) (cid:3) ln n xn ! . (5.12)If f satisfies the conditions of Taylor,s Theorem in [ a, ah ] and x is any point of [ a, ah ]then it also satisfies the conditions in the interval [ a, x ] . Then replacing ah by x or h by -Calculus 21 x/a in 5.10, we get another form of Taylor’s product as follows: f ( x ) = f ( a ) . (cid:2) f [1] ( a ) (cid:3) ln( xa ) . (cid:2) f [2] ( a ) (cid:3) ln2( xa )2! ... (cid:2) f [ n ] ( a ) (cid:3) ln n ( xa ) n ! ... = Π ∞ n =0 (cid:2) f [ n ] ( a ) (cid:3) ln n ( xa ) n ! . (5.13)6. Some applications of G -calculus Expansion of some useful functions in Taylor’s product. (i) With the help of geometric Taylor’s series, we can express different functions asa product of different functions. For, let f ( x ) = e x . Then f [1] ( x ) = f [2] ( x ) = f [3] ( x ) = ..... = e x . Hence f (1) = f [1] (1) = f [2] (1) = f [3] (1) = ..... = e. Thereforefrom (5.12) e x = e.e ln x .e ln2 x .e ln3 x ..... = e x + ln2 x + ln3 x + ... (ii) Let f ( x ) = sin( x ) We can approximate the value of f ( x ) at different points,say at x = π . In the figure 3, we have given a comparison of first order linearapproximation and first order exponential approximation with the help of andgeometric Taylor’s series respectively.By ordinary Taylor’s series, first order linear approximation is given by L ( x ) = f ( π x − π f ′ ( π π x − π π L ( x ) = 12 + ( x − π √ . By geometric Taylor’s series, first order exponential approximation is given by E ( x ) = f ( π . h f [1] ( π i ln ( xπ/ )= sin( π . (cid:2) e π cot( π ) (cid:3) ln ( xπ )i.e. E ( x ) = 12 . h e π √ i ln ( xπ ) . For the graphical approximation, we have made the Table 1 for sin( x ) , L ( x ) and E ( x ) , then plotting the values we get the figure 3. From the FIGURE 3, it is clearthat geometric Taylor’s series gives better approximated value of the function f ( x ) = sin( x ) at x = π than Taylor’s approximation given by Michael Coco in [8]with the help of multiplicative derivative f ∗ ( x ) = lim h → (cid:20) f ( x + h ) f ( x ) (cid:21) h . Table 1.
Approximation at x = π x. sin(x) L(x) E(x)-2 -0.9093 -1.6855 --1.6 -0.99957 -1.33909 --1.2 -0.93204 -0.99268 --0.8 -0.71736 -0.64627 --0.4 -0.38942 -0.29986 -0 0 0.04655 -0.4 0.389418 0.39296 0.4310210.8 0.717356 0.73937 0.6316331.2 0.932039 1.085781 0.7898581.6 0.999574 1.432191 0.9256172 0.909297 1.778601 1.0467932.4 0.675463 2.125011 1.1574862.8 0.334988 2.471421 1.2601593.2 -0.05837 2.817831 1.3564323.6 -0.44252 3.164242 1.4474384 -0.7568 3.510652 1.5340074.4 -0.9516 3.857062 1.616774.8 -0.99616 4.203472 1.696225.2 -0.88345 4.549882 1.77275 -1012-2 -1 0 1 2 3 4 5sin(x) L(x) E(x) Figure 3.
Exponential Approximation(iii) G -derivative gives total growth of a growth function. For, let y = a.b x , where a =initial amount > b = growth(or decay) factor, x =time and y =total amountafter time period x. Then, d G ydx G = b x , which is the total growth or total decayaccording to b > < b < -Calculus 23 (iv) It is easy to find ordinary derivative of complicated product or quotient functionswith the help of G -derivative. For let, f ( x ) = e − /x x n sin x . Then f G ( x ) = d G dx G ( e − /x ) d G dx G ( x n ) . d G dx G (sin x ) = e /x e n .e x cot x = e x − n − x cot x Therefore ordinary derivative is given by f ′ ( x ) = f ( x ) ln (cid:0) f G ( x ) (cid:1) x = e − /x x n +1 sin x (cid:18) x − n − x cot x (cid:19) . (v) (Price Elasticity) With the aid of G-derivative, we can find price elasticity topredict the impact of price changes on unit sales and to guide the firm’s profit-maximizing pricing decisions. According to [15](page no. 83), the price elasticityof demand is the ratio of the percentage change in quantity and the percentagechange in the good’s price, all other factors held constant. If x and y representsprice and quantity respectively, then the price elasticity E p is given by E p = % change in y % change in x = ∆ y/y ∆ x/x = x ∆ y ∆ x y If price change is very small to the initially considered price, then making ∆ x → , we get E p = x y ′ y = ln (cid:18) e xy ′ y (cid:19) = ln( y [1] )or y [1] = e E p . where y [1] is the G -derivative of y. Thus, natural logarithm of G -derivative givesthe price elasticity. In other words we can say that, G -derivative of quantity withrespect to the price is the exponential price elasticity. We know thatResiliency = e (elasticity) = e E p . Therefore, G -derivative gives the resiliency.7. Acknowledgment
It is pleasure to thank Prof. M. Grossman and Prof. Jane Grossman for their construc-tive suggestions and inspiring comments regarding the improvement of the G -calculus. Conclusion
Based on the work of Grossman and Katz’s [11] and Grossman [10], we studied some re-sults on bigeometric calculus in our paper [5], here we have discussed more about the saidtopic. In geometric calculus, Grossman and Katz took ordinary sum(+) to produce incre-ment to the independent variable x such as x , x + h, x +2 h, ... In that case some problemarise to discuss independently about the geometric arithmetic system ( ⊕ , ⊖ , ⊙ , ⊘ ) . In G -calculus, geometric sum( ⊕ ) is taken to produce increment to the independent variable x such as x , x ⊕ h, x ⊕ e ⊙ h, ...(equivalently a, ah, ah , ah , ... ). Instead of mixing theordinary arithmetic system(+ , − , × , ÷ ) and geometric arithmetic system ( ⊕ , ⊖ , ⊙ , ⊘ ), weare trying to formulate basic identities independently. As well as in [4], here, we are tryingto bring up researchers’ attention to G -calculus and its applications to different branchesof analysis. Advantages of G -calculus will be apparent when it becomes useful in differentpractical fields namely finance, economics, statistics etc. References [1] A.E. Bashirov, M. Rıza,
On Complex multiplicative differentiation , TWMS J. App. Eng. Math.1(1)(2011) 75-85.[2] A. E. Bashirov, E. Mısırlı, Y. Tandoˇgdu, A. ¨Ozyapıcı,
On modeling with multiplicative differentialequations , Appl. Math. J. Chinese Univ. 26(4)(2011) 425-438.[3] A. E. Bashirov, E. M. Kurpınar, A. ¨Ozyapici,
Multiplicative Calculus and its applications , J. Math.Anal. Appl. 337(2008) 36-48.[4] Khirod Boruah and Bipan Hazarika,
Application of Geometric Calculus in Numerical Analysis andDifference Sequence Spaces , arXiv:1603.09479v1, 31 May 2016.[5] Khirod Boruah and Bipan Hazarika,
Some basic properties of G-Calculus and its applications innumerical analysis , arXiv:1607.07749v1, 24 July 2016.[6] A. F. C¸ akmak, F. Ba¸sar,
On Classical sequence spaces and non-Newtonian calculus , J. Inequal. Appl.2012, Art. ID 932734, 12pp.[7] Duff Campbell,
Multiplicative Calculus and Student Projects , Department of Mathematical Sciences,United States Military Academy, West Point, NY,10996, USA.[8] Michael Coco,
Multiplicative Calculus , Lynchburg College.[9] M. Grossman, R. Katz,
Non-Newtonian Calculus , Lee Press, Piegon Cove, Massachusetts, 1972.[10] M. Grossman,
Bigeometric Calculus: A System with a scale-Free Derivative , Archimedes Foundation,Massachusetts, 1983.[11] M. Grossman,
An Introduction to non-Newtonian calculus , Int. J. Math. Educ. Sci. Technol.10(4)(1979) 525-528.[12] Jane Grossman, M. Grossman, R. Katz,
The First Systems of Weighted Differential and IntegralCalculus , University of Michigan, 1981.[13] Jane Grossman,
Meta-Calculus: Differential and Integral , University of Michigan, 1981.[14] U. Kadak and Muharrem ¨Ozl¨uk,
Generalized Runge-Kutta method with respect to non-Newtoniancalculus , Abst. Appl. Anal., Vol. 2015 (2015), Article ID 594685, 10 pages.[15] W.F. Samuelson, S.G. Mark,
Managerial Economics , Seventh Edition, 2012.[16] D. Stanley,
A multiplicative calculus , Primus IX 4 (1999) 310-326.[17] S. Tekin, F. Ba¸sar,
Certain Sequence spaces over the non-Newtonian complex field , Abstr. Appl.Anal. 2013. Article ID 739319, 11 pages.[18] Cengiz T¨urkmen and F. Ba¸sar,
Some Basic Results on the sets of Sequences with Geometric Calculus ,Commun. Fac. Fci. Univ. Ank. Series A1. Vol G1. No 2(2012) 17-34.[19] A. Uzer,