Method of Continuous Representation of Functions Having Real and Imaginary Parts

www.sjmmf.org Journal of Modern Mathematics Frontier, Volume 5 2016 doi: 10.14355/jmmf.2016.05.002

Method of Continuous Representation of Functions Having Real and Imaginary Parts Ivanov K.S. *1 Department of Aero‐Space Control Systems, Almaty University of Power Engineering and Telecommunication, 050013. Kazakhstan, Almaty, Baytursinov street, 126 *1ivanovgreek@mail.ru

Abstract Method of graphic representation of function which accepts real and imaginary values in the full range of values of argument is presented. Imaginary values of function are represented by complex numbers. The method axes contain two mutually perpendicular real axes and imaginary axis which is perpendicular to the real axes. The real part of a complex number is postponed on horizontal real axis. The imaginary part of complex number is postponed on imaginary axis. The real part of function is represented in vertical real plane. The imaginary part of function is represented in horizontal imaginary plane. The method allows building the full graphic representation of function within the full range of argument change. By means of a method new properties are found.

Keywords Complex Number; Real Plane, Imaginary Plane; Full Image

Introduction It is known that the complex number m  a  ib can be presented in the representation of a point M (a, ib) on the imaginary plane having the real axis and imaginary axis. The parametre a is postponed on the real axis and the parametre ib (an imaginary part) is postponed on the imaginary axis in a perpendicular direction.

In mathematics the cases take place when at the real values of argument x the function y  f ( x) accepts imaginary values in the representation of an imaginary number having only an imaginary part. Earlier the attempts of a graphical representation of an imaginary part of function (circle) in a pseudo‐Euclidean plane in the representation of two open branches of a hyperbola [1] were undertaken. However the real part of function has not been represented.

The task in view to create a method which allows to represent completely graphically a function y  f ( x) which accepts the real and imaginary values in a full continuous range of argument change and to analyse its a reality compliancy.

Method of Continuous Representation of Functions The method of continuous representation of function y  f ( x) consists in the following (fig. 1).

The method coordinate system has real axes Ox , Oy and imaginary axis Oiy which is placed perpendicularly to the real axes. Here i   1 .

y

M(x, y)

iy Mi(x, iy)

O

x

FIG. 1. GRAPHICAL REPRESENTATION OF THE REAL AND IMAGINARY POINTS

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Argument x has only the real values in a full continuous range of values   x   and is postponed on axis Ox .

Function y  f ( x) can have the real or imaginary values. Real values y are postponed on axis Oy , imaginary values iy are postponed on axis Oiy . Real point M lays in plane Oxy , imaginary point M i lays in plane Oxiy . The general plot of function will contain the real and imaginary parts.

Graphical Representation of Functions Circle Function of a circle with radius r looks like

y  r 2  x 2 . (1)

y

iy y=ix

r OF

x

FIG. 2. GENERAL PLOT OF CIRCLE

At r  x  r real function (circle) occurs in the real (vertical) plane Oxy .

At   x  r and r  x   imaginary function (equiangular hyperbola) occurs in an imaginary (horizontal) plane Oxiy .

The full plot of a circle containing the real part and imaginary parts (shown by a dot line) is presented in fig. 2.

The dotted straight lines which are drawing in an imaginary plane under angle 45° to axis Ox are representing equiangular hyperbola asymptotes.

It would be more evident to represent an imaginary part of function in the form of a real‐valued function in a conditional real plane.

The imaginary part of function can be conditionally presented in the representation of a real‐valued function according to the following theorem.

The theorem of transformation of functions.

The imaginary part of function can be presented as conditional real function and on the contrary by multiplication of the equation of function by imaginary number i .

Generally function looks like y 

A  x where A ‐ any positive number.

At x  A imaginary function takes place.

Letʹs multiply the function equation by i . We will gain iy  i A  x . We will mark out iy  z where z ‐ the designation of a conditional real number in conditionally real plane Oxz . After introduction i under a root we

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will gain z  x  A . At x  A the gained equation expresses a conditionally real function in plane Oxz as was to be shown.

For the circle equation it is had y  r 2  x 2 . After transformation with multiplication by i we will gain the equation of the real equiangular hyperbola in system Oxz or an imaginary equiangular hyperbola in system Oxiy z  x 2  r 2 . (2)

Check of Adequacy of the Accepted Technique of Graphic Representation to Mathematical Reality Check can be executed on the basis of the following statement pseudo‐Euclidean geometry: the isotropic straight progressing through focus by a curve, is a tangent to this curve [2]. Such statement does not conform to a usual graphical representation according to which any 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 y  ix . According to the accepted technique this straight must be in imaginary plane Oxiy . Therefore the isotropic straight can be a tangent to an imaginary part of the circle which is placed in an imaginary plane. We will present the analytical 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 (imaginary circle).

Letʹs determine co‐ordinates of intersection points of some straight

y  kx with angular coefficient k which is

passing through origin of co‐ordinates and circle (1). We will carry out a joint resolution of the equation of straight and the circle equation.

Letʹs substitute value

y from the equation of a direct circle in the equation.

kx  r 2  x 2 . After squaring we will gain k 2 x 2  r 2  x 2 . Further

(k 2  1) x 2  r 2 . (3) For isotropic straight k  i . From the equation (3) we will find values x at k  i x1, 2   r 2 / (1  1) . (4)

From here we will find abscissas of intersection points x1   , x 2   .

Letʹs substitute these values in the equation (1). We will gain ordinates of intersection points in an imaginary plane y1  k , y2   k  .

Thus each straight matching to an isotropic straight has two conterminous intersection points with an imaginary circle at x1   и x2   . Hence, each straight is a tangent to an imaginary part of circle (that is to a hyperbola).

Analytical regularity matches to a graphic representation. The isotropic straight coincides with asymptotes of an imaginary equiangular hyperbola which have points of contact with a curve in infinity at x1   , x2   .

Letʹs consider another way 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 1 ( x1 y1 ) looks like

xx1  yy1  r 2 . (5) Letʹs substitute values of co‐ordinates of point M 1 ( x1 , y1 ) which is on an isotropic straight at y1  ix1 , we will gain xx1  yix1  r 2 or 16

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x  iy  r 2 / x1 . (6) At x1   we will gain the equation of the tangent x  iy which is passing through point M 1 . We will multiply this equation on i , we will gain

y  ix . (7) The equation (7) is the equation of an isotropic straight.

Further we will substitute values of co‐ordinates of point M 2 ( x2 , y2 ) which is on an isotropic straight at y2  ix2 in the equation (5), we will gain xx2  yix2  r 2 or x  iy  r / x 2 . (8) 2

At x2   we will gain the equation of the tangent which is passing through point M 2 y  ix . This equation coincides with the equation (7) and also is the equation of an isotropic straight.

Thus graphical acknowledging of known regularity pseudo‐Euclidean geometry is gained.

Here it is possible to note surprising regularity.

As is known contact point M of straight and curve of the second order is a point in which two infinitely close intersection points M 1 (, i) and M 2 (,  i) are placed. In the considered case the coincidence of two points M 1 and M 2 in one point means that the isotropic straight (7) becomes closed in one imaginary point M which is

simultaneously 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 necessary to note that the equation (7) defines two mutually perpendicular isotropic straight lines.

This regularity can be tied up to an explanation of infinity and reticence of the Universe.

Ellipse Function of ellipse looks like y  (b / a) a 2  x 2 . (9)

Here a and b ‐ big and small semi‐axes of an ellipse.

The full plot of ellipse with the image only the right imaginary part is presented in fig. 3.

The real part of ellipse is presented on vertical plane Oxy . The imaginary part of ellipse is presented on horizontal plane. The imaginary part of function can be converted to a real‐valued function.

Letʹs multiply the equation (9) by i , we will gain

iy  (b / a ) x 2  a 2 . (10) Here a  x  a .

It is possible to present the equation (10) in the form z  (b / a ) x 2  a 2 ‐ the equation of the real hyperbola in system

Oxz or the imaginary hyperbola in system Oxiy .

Letʹs execute check of reliability of the found regularity and coincidence of the accepted technique with mathematical reality. We will check a condition of contact of the isotropic straight passing through the ellipse pole F with the imaginary hyperbola. For this purpose we will find two intersection points of an isotropic straight with imaginary hyperbola. 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 (c, 0) looks like

y  i ( x  c) . (11)

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Here c  a 2  b 2 .

y

y=i(x-c)

iy M

b O

a

F

x

FIG. 3. FULL PLOT OF ELLIPSE WITH THE RIGHT IMAGINARY PART

Letʹs substitute value y from the equation (11) in the ellipse equation (10), we will gain

i ( x  c)  (b / a) a 2  x 2 . (12) Letʹs erect the equation (12) in square and after conversions we will gain c 2 x 2 / a 2  2cx  a 2  0 . (13)

The solution of the equation (13) looks like x1,2  a 2 / c . (14) Formula (14) asserts that the point of contact of isotropic straight and ellipse in its imaginary 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 x1,2   . It confirms earlier inferred regularity for a circle and a tangent in its imaginary part.

Thus graphical acknowledging of known regularity of pseudo‐Euclidean geometry is gained.

Adequacy of the accepted way of geometrical representation with a mathematical reality is proved.

Convertibility of the Real and Imaginary Parts of Function According to the theorem of transformation of functions proved above the imaginary part of function can be presented in the form of conditional real function and on the contrary by multiplication of the equation of function by imaginary number i .

y y=-(x-с) b O

F(c, 0) a

x

-45°

FIG. 4. TANGENT TO ELLIPSE PASSING FROM POLE OF IMAGINARY HYPERBOLA UNDER ANGLE "  / 4"

This regularity defines property of convertibility of the real and imaginary parts of function. On the basis of 18

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convertibility of function it is possible to change full geometrical representation of a curve by transformation of the real part in imaginary part and on the contrary.

Property of convertibility of functions allows formulating brand new regularity: real straight with angular coefficient k  1 (or k  1 ) passing from imaginary 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 regularity, for example, for an ellipse with a small semi‐axis b (fig. 4). A straight passing under angle "  / 4" through point F (c, 0) tangent to ellipse is. Here F (c, 0) ‐ focus of the imaginary hyperbola, c  2b .

Isolated Point The graphical representation of a curve taking into account its real and imaginary part allows bringing in correction to the description of some theoretical regularity.

y

-1

iy

M 1

1 x

0

-1

FIG. 5. ISOLATED POINT FROM REAL PART OF CURVE

For example, it is possible to correct concept ʺisolated pointʺ [3 ‐ p. 244]. According to [3] isolated point M (0, 0) has a place in curve y 2  x 4  x 2 . The isolated point does not lay on the main curve of the equation (fig. 5).

In fig. 5 considered curve is presented by contour line. We will build an imaginary part of this curve in an imaginary plane x0iy ‐ it is shown by a dot line. The considered curve passes in an imaginary part on intervals 1  x  0, 0  x  1 .

Hence the point M (0, 0) is not isolated point. It lays on the curve having the real and imaginary parts.

Conclusions The new method of a graphical representation of the functions, allowing gaining a composite image of the function containing the real and imaginary parts is offered. It is proved that the offered method reflects real geometry of analytical regularity. Known analytical regularity of contact of isotropic straight with investigated curve has gained a graphical form.

Convertibility of the real and imaginary parts of function is proved. New property is found: the real straight with angular coefficient, equal «+1» or «‐1» passing from imaginary curve pole is tangent to curve real part. The concept of an isolated point is corrected.

The executed researches allow graphical presenting of analytic geometry in the real and imaginary planes. The method of continuous representation of functions can be used for the subsequent research of functions. The analytic geometry in an imaginary plane supplements and generalizes the mathematical description of functions on a plane. Parabola research at which the imaginary part will appear asymmetrical in relation to the real part is of interest.

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Creation of method of the general graphical representation of functions in space with the description of infinity and reticence of space is possible in the long term.

REFERENCES

[1]

Laptev G.F. Elements of vector calculus. М: Science. (1975). 335 p.

[2]

Walter Noll. Euclidean geometry and Minkowskian chronometry. American Mathematical Monthly. 71:129‐44. (1964).

[3]

Christopher Clapham and James Nicholson. Oxford Concise Dictionary of Mathematics. Fourth Edition. (2009). 510 p.

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