Method of Continuous Representation of Functions Having Real and Imaginary Parts

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1 Journal of Modern Mathematics Frontier, Volume 5 06 doi: /jmmf Method of Continuous Representation of Functions Having Real and Imaginar Parts Ivanov K.S. * Department of Aero Space Control Sstems, Almat Universit of Power Engineering and Telecommunication, Kazakhstan, Almat, Batursinov street, 6 * ivanovgreek@mail.ru Abstract Method of graphic representation of function which accepts real and imaginar values in the full range of values of argument is presented. Imaginar values of function are represented b comple numbers. The method aes contain two mutuall perpendicular real aes and imaginar ais which is perpendicular to the real aes. The real part of a comple number is postponed on horizontal real ais. The imaginar part of comple number is postponed on imaginar ais. The real part of function is represented in vertical real plane. The imaginar part of function is represented in horizontal imaginar plane. The method allows building the full graphic representation of function within the full range of argument change. B means of a method new properties are found. Kewords Comple Number; Real Plane, Imaginar Plane; Full Image Introduction It is known that the comple number m a ib can be presented in the representation of a point M ( aib, ) on the imaginar plane having the real ais and imaginar ais. The parametre a is postponed on the real ais and the parametre ib (an imaginar part) is postponed on the imaginar ais in a perpendicular direction. In mathematics the cases take place when at the real values of argument the function f( ) accepts imaginar values in the representation of an imaginar number having onl an imaginar part. Earlier the attempts of a graphical representation of an imaginar part of function (circle) in a pseudo Euclidean plane in the representation of two open branches of a hperbola [] were undertaken. However the real part of function has not been represented. The task in view to create a method which allows to represent completel graphicall a function f( ) which accepts the real and imaginar values in a full continuous range of argument change and to analse its a realit complianc. Method of Continuous Representation of Functions The method of continuous representation of function f( ) consists in the following (fig. ). The method coordinate sstem has real aes O, O and imaginar ais Oi which is placed perpendicularl to the real aes. Here i. M(, ) i M i (, i) O FIG.. GRAPHICAL REPRESENTATION OF THE REAL AND IMAGINARY POINTS 4

2 Journal of Modern Mathematics Frontier, Volume Argument has onl the real values in a full continuous range of values and is postponed on ais O. Function f( ) can have the real or imaginar values. Real values are postponed on ais O, imaginar values i are postponed on ais Oi. Real point M las in plane O, imaginar point M i las in plane Oi. The general plot of function will contain the real and imaginar parts. Graphical Representation of Functions Circle Function of a circle with radius r looks like r. () r i =i O F FIG.. GENERAL PLOT OF CIRCLE At r r real function (circle) occurs in the real (vertical) plane O. At r and r imaginar function (equiangular hperbola) occurs in an imaginar (horizontal) plane Oi. The full plot of a circle containing the real part and imaginar parts (shown b a dot line) is presented in fig.. The dotted straight lines which are drawing in an imaginar plane under angle 45 to ais O are representing equiangular hperbola asmptotes. It would be more evident to represent an imaginar part of function in the form of a real valued function in a conditional real plane. The imaginar part of function can be conditionall presented in the representation of a real valued function according to the following theorem. The theorem of transformation of functions. The imaginar part of function can be presented as conditional real function and on the contrar b multiplication of the equation of function b imaginar numberi. Generall function looks like A where A an positive number. At A imaginar function takes place. Letʹs multipl the function equation b i. We will gain i i A. We will mark out i z where z the designation of a conditional real number in conditionall real plane Oz. After introduction i under a root we 5

3 Journal of Modern Mathematics Frontier, Volume 5 06 will gain z A. At A the gained equation epresses a conditionall real function in plane Oz as was to be shown. For the circle equation it is had r. After transformation with multiplication b i we will gain the equation of the real equiangular hperbola in sstem Oz or an imaginar equiangular hperbola in sstem Oi z r. () Check of Adequac of the Accepted Technique of Graphic Representation to Mathematical Realit Check can be eecuted on the basis of the following statement pseudo Euclidean geometr: the isotropic straight progressing through focus b a curve, is a tangent to this curve []. Such statement does not conform to a usual graphical representation according to which an straight which is passing through a pole of a curve must cross a curve. However the method of continuous representation of functions is capable to give evident geometrical representation of this phenomenon. The isotropic isotropic line which is passing through focus of circle F (or through the circle centre) looks like i. According to the accepted technique this straight must be in imaginar plane Oi. Therefore the isotropic straight can be a tangent to an imaginar part of the circle which is placed in an imaginar plane. We will present the analtical description of this phenomenon. The tangent equation to the curve of the second order which is passing through the set point can be gained from condition of coincidence of two intersection points of a straight with curve (imaginar circle). Letʹs determine co ordinates of intersection points of some straight k with angular coefficient k which is passing through origin of co ordinates and circle (). We will carr out a joint resolution of the equation of straight and the circle equation. Letʹs substitute value from the equation of a direct circle in the equation. k r. After squaring we will gain For isotropic straight k r. Further k i. From the equation (3) we will find values at ( k ) r. (3), r /( ) From here we will find abscissas of intersection points,. k i. (4) Letʹs substitute these values in the equation (). We will gain ordinates of intersection points in an imaginar plane k, k. Thus each straight matching to an isotropic straight has two conterminous intersection points with an imaginar circle at и. Hence, each straight is a tangent to an imaginar part of circle (that is to a hperbola). Analtical regularit matches to a graphic representation. The isotropic straight coincides with asmptotes of an imaginar equiangular hperbola which have points of contact with a curve in infinit at,. Letʹs consider another wa of statement that the isotropic straight which is through passing through focus of a circle is tangent to circle. The tangent equation to a circle which is passing through point of contact M( ) looks like r. (5) Letʹs substitute values of co ordinates of point M(, ) which is on an isotropic straight at i i r or, we will gain 6

4 Journal of Modern Mathematics Frontier, Volume i r /. (6) At we will gain the equation of the tangent i which is passing through point M. We will multipl this equation on i, we will gain The equation (7) is the equation of an isotropic straight. i. (7) Further we will substitute values of co ordinates of point M (, ) which is on an isotropic straight at i in the equation (5), we will gain i r or i r /. (8) At we will gain the equation of the tangent which is passing through point M coincides with the equation (7) and also is the equation of an isotropic straight. Thus graphical acknowledging of known regularit pseudo Euclidean geometr is gained. Here it is possible to note surprising regularit. i. This equation As is known contact point M of straight and curve of the second order is a point in which two infinitel close intersection points M (, i ) and M (, i ) are placed. In the considered case the coincidence of two points M and M in one point means that the isotropic straight (7) becomes closed in one imaginar point M which is simultaneousl in positive and negative infinite. The point which moves on a straight to positive infinite appears in negative infinite. At the same time it is necessar to note that the equation (7) defines two mutuall perpendicular isotropic straight lines. This regularit can be tied up to an eplanation of infinit and reticence of the Universe. Ellipse Function of ellipse looks like Here a and b big and small semi aes of an ellipse. ( / ). (9) b a a The full plot of ellipse with the image onl the right imaginar part is presented in fig. 3. The real part of ellipse is presented on vertical plane O. The imaginar part of ellipse is presented on horizontal plane. The imaginar part of function can be converted to a real valued function. Letʹs multipl the equation (9) b i, we will gain Here a a. i b a a ( / ). (0) It is possible to present the equation (0) in the form Oz or the imaginar hperbola in sstem Oi. ( / ) the equation of the real hperbola in sstem z b a a Letʹs eecute check of reliabilit of the found regularit and coincidence of the accepted technique with mathematical realit. We will check a condition of contact of the isotropic straight passing through the ellipse pole F with the imaginar hperbola. For this purpose we will find two intersection points of an isotropic straight with imaginar hperbola. These two points should coincide with a contact point. The equation of the isotropic straight which is passing through the right focus of ellipse F(,0) c looks like i( c). () 7

5 Journal of Modern Mathematics Frontier, Volume 5 06 Here c a b. i =i(-c) M b O a F FIG. 3. FULL PLOT OF ELLIPSE WITH THE RIGHT IMAGINARY PART Letʹs substitute value from the equation () in the ellipse equation (0), we will gain i ( c) ( b/ a) a Letʹs erect the equation () in square and after conversions we will gain. () The solution of the equation (3) looks like c / a c a 0. (3) a c. (4), / Formula (4) asserts that the point of contact of isotropic straight and ellipse in its imaginar part occurs. In this point two intersection points of straight and ellipse coincide. Letʹs note also that in that specific case at c 0 ellipse is transforming in the circle with r a and abscissa of contact point,. It confirms earlier inferred regularit for a circle and a tangent in its imaginar part. Thus graphical acknowledging of known regularit of pseudo Euclidean geometr is gained. Adequac of the accepted wa of geometrical representation with a mathematical realit is proved. Convertibilit of the Real and Imaginar Parts of Function According to the theorem of transformation of functions proved above the imaginar part of function can be presented in the form of conditional real function and on the contrar b multiplication of the equation of function b imaginar numberi. =-(-с) b O a F(c, 0) -45 FIG. 4. TANGENT TO ELLIPSE PASSING FROM POLE OF IMAGINARY HYPERBOLA UNDER ANGLE " /4" This regularit defines propert of convertibilit of the real and imaginar parts of function. On the basis of 8

6 Journal of Modern Mathematics Frontier, Volume convertibilit of function it is possible to change full geometrical representation of a curve b transformation of the real part in imaginar part and on the contrar. Propert of convertibilit of functions allows formulating brand new regularit: real straight with angular coefficient k (or k ) passing from imaginar curve pole is a tangent to the real curve. On the basis of this statement it is possible to represent in the real plane brand new regularit, for eample, for an ellipse with a small semi ais b (fig. 4). A straight passing under angle " /4" through point F(,0) c tangent to ellipse is. Here F(,0) c focus of the imaginar hperbola, c b. Isolated Point The graphical representation of a curve taking into account its real and imaginar part allows bringing in correction to the description of some theoretical regularit. i - M 0 - FIG. 5. ISOLATED POINT FROM REAL PART OF CURVE For eample, it is possible to correct concept ʺisolated pointʺ [3 p. 44]. According to [3] isolated point M (0, 0) has a place in curve 4. The isolated point does not la on the main curve of the equation (fig. 5). In fig. 5 considered curve is presented b contour line. We will build an imaginar part of this curve in an imaginar plane 0i it is shown b a dot line. The considered curve passes in an imaginar part on intervals 0, 0. Hence the point M (0, 0) is not isolated point. It las on the curve having the real and imaginar parts. Conclusions The new method of a graphical representation of the functions, allowing gaining a composite image of the function containing the real and imaginar parts is offered. It is proved that the offered method reflects real geometr of analtical regularit. Known analtical regularit of contact of isotropic straight with investigated curve has gained a graphical form. Convertibilit of the real and imaginar parts of function is proved. New propert is found: the real straight with angular coefficient, equal «+» or passing from imaginar curve pole is tangent to curve real part. The concept of an isolated point is corrected. The eecuted researches allow graphical presenting of analtic geometr in the real and imaginar planes. The method of continuous representation of functions can be used for the subsequent research of functions. The analtic geometr in an imaginar plane supplements and generalizes the mathematical description of functions on a plane. Parabola research at which the imaginar part will appear asmmetrical in relation to the real part is of interest. 9

7 Journal of Modern Mathematics Frontier, Volume 5 06 Creation of method of the general graphical representation of functions in space with the description of infinit and reticence of space is possible in the long term. REFERENCES [] Laptev G.F. Elements of vector calculus. М: Science. (975). 335 p. [] Walter Noll. Euclidean geometr and Minkowskian chronometr. American Mathematical Monthl. 7:9 44. (964). [3] Christopher Clapham and James Nicholson. Oford Concise Dictionar of Mathematics. Fourth Edition. (009). 50 p. 0

THE INVERSE GRAPH. Finding the equation of the inverse. What is a function? LESSON

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