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The Function Magna

 

The function magna

Consider a straight line in positive x direction originating from point M where a circle of unit radius centred at the origin O of the Cartesian Coornite plane, cuts the positive axis. The distance x at a point L on this line from its initial point M is the argument of the function Magna. If from any point of this llne, there is drawn a straight line joining the point L with the origin O, (fig 1) the angle theta made by the line OP with the positive x-axis the required value of the function magna against its argument x.  theta=mag(x).

Figure (1)

A line drawn perpendicularly to the x-axis from the point Q meeting it at Q’, will have the measure of the sine of the angle theta. Let QQ’ be denoted by y.

Hence,  

y=sintheta                                                                                                                              theta=arcsiny

Fig(2)



The value of every angle against any value of x can be found out. Thus for x=1, QQ’ or y is .707 and theta=arcsiny. If on fig 2, several values of the sines of the angles against the values on the perpendicular line initiating from the point P; such as, 0.5 and others are measured, the table showing real numbers, and the values of the sines of the angles related to them by the function magna, can be constructed easily. Moreover, the angles these numbers are sines of, can be determined by substituting them in the functional series approximating the function arcsintheta and then evaluating that series.  

theta or mag(x)= arcsiny where y=measure of the line segment, QQ’, the sine of the angle to be found out

mag(x)=y+y3/6+3y5/40+------    (1)                                                                                       as  mag(0.5)=0.45+(0.45)3/6+3(0.45)5 /40+------                                                                                     =0.45+ 0.0151+0.0013+-------                                                                             or mag(.5) =0.4664                                                                                                                             

                                                                                                                                      

Table 1

            x                             siny                                           y or magx  

          0.1                            0.1                                              0.1

          0.2                            0.2                                              0.2     

          0.3                            0.35                                            0.35714  

          0.5                            0.45                                            0.46518                                           

          0.6                            0.50                                            0.52083

          0.8                            0.60                                            0.636                              

          1.0                            0.70                                            0.76

          1.5                            0.85                                            0.95

          2.0                            0.90                                            1.02

          2.5                            0.93                                            1.06

          3.0                            0.95                                            1.09

          3.5

          4.0

The only way is to find a polynomial series approximating mag(x) near a specific constant say ‘a’. Based on the theorem of Brook Taylor, the function has a polynomial approximation given as:.   

f(x)=f(a)+(x-a)f’(a)+(x-a)2/2f”(a)+----------                                  (2)

For the values of x between 0 and 3. According to a well known theorem of elementry calculus, as x tends to 0, sinx tends to x. Thefore; if siny is very small, y will be nearly equal to siny which is nearly the same as x in the figure. Hence y=x from 0 to 0.3.

Next, in the open interval (0.3, 1), the terms f’(a) and f”(a) are, in fact, the rates of change of f of the first and second order respectively. The rate of change or derivative of first order of f is unknown throughout the domain of f, the sole method of knowing it is taking a small increment in the value of x, denoted by delx, and then finding a small change in the value of theta against that change in x, denoted deltheta. The ratio deltheta/delx at any point,c, is the first order derivative, that is, deltheta/delx=f’(c).

In the interval (.3,1) the difference between f(.6) and f(.5) may be thought to be the average over (.3,1); hence, dividing f(.6)--f(.5) by .1 gives the value of f’(a) approximatey.

Thus      f’(a)=(0.52083—0.46518)/0.1

         or  f’(a)= 5565/10000 or 55/100

         or  f’(a)=11/20

                                                                                                                                               The rate of change of second order, f’’(x) which is, in fact, the rate of change of rate of change of f, will be found by dividing the rates of change of first order at two successive points any where over its domain.  

f’’(x)=f’(c2)/f’(c1)                                                                                                                     where c1 and c2 are two consecutive points in the domain of f. An attempt to find this ratio reveals that to be neglible, so the value of f’’ can be safely taken as 1 here.

Evaluating in (2) for f’(a) and f”(a) the approximating polynomial for f in the interval (0.3, 1) around 0.5 is got as follows

f(x)=0.465+(x-0.5)11/20+(x-0.5)2/2

      =47/100+(x-1/2)11/20+(x-1/2)2/2

      =47/100+(2x-1/2)/(11/20)+x2/2-x/2+1/8

      =47/100+(22x-11)/40+(4x2-4x+1)/8

      =47/100+(2x-6+20x2)/40

      =32/100+x/20-x2/2

Hence near 0.5, 

f(x)=1/3+x/20+x2/2

By conjecturing about the rate of change of f near 1.5, its most proper value is 1/7.

So, substitung for f(a) and f’’(a) in (2)

f(x)=95/100+((x-3)/2))1/7+(x-3/2)2/2

      =17/20+(2x-3)/14+(4x2-12x+9)/8

      =17/20+(8x-12+28x2-84x+63)/56

      =17/20+51/56-(76x+28x2)/56

      =99/56-(19/14)x+x2/2

Thus, near 1.5  

f(x)=7/4-(19/14)x+x2/2

Once again, taking the most appropriate value for the rate of change of f near 2.5 to be 1/10

f(x)=53/50+((2x-5)/2))/10+(4x2-20x+25)/8

      =53/50+(2x-5)20+(4x2-20x+25)/8

      = 53/50+(4x-10+20x2-100x+125)/40

      =53/50+115/40-(96x)/40+(20x2)/40

      =157/40-12x/5+x2/2

Thus, near 2.5

f(x)=157/40-12x/5+x2/2

Upon examining table (1), it can be observed that the values of the function mag(x) form a sequence that may be rgarded as infinite and a closer view of it will suggest that it is Cauchy and most probabally having the limit lying between pi/3 and pi/2.

0.76, 1.02, 1.09, 1.12, 1.136, 1.141, 1.146, 1.149, 1.502,..……………..

Theorem: If x approaches infinity, the limit of the function mag(x) lies somewhere between pi/3 and pi/2.

  

 

 

 

 

 

 

tth

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