<p> hatching utility </p>

80db06c9-cf43-4529-83b7-d2839ba82f14 26 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Erzeugt eine Matrix für die Darstellung vertikaler Linien zwischen den Funktionen f(x) und 
g(x) im Bereich von x1 bis x2 mit 2*n Linien. Generates a matrix for 2*N vertical lines (hatching) 
between functions f(x) and g(x) in x-range from x1 to x2 f g x.1 x.2 N hatch dx x.2 x.1 - N / (: h x.1 x.1 f x.1 x.1 g 22 mat : x x.1 x.2 dx - x.1 dx + range h h x dx 2 / + x dx 2 / + g x dx 2 / + x dx 2 / + f x dx + x dx + f x dx + x dx + g 42 mat stack : for h 41 line :

click on + to expand the collapsed area. click on - to collapse again.

x f 1 x / :

x g 0 :

b 3 :

The quadratic hyperbola 1/x that generates ln(x)

x f x g x f x g 1 b 10 hatch x f x g 0.1 1 10 hatch x ln 5 1 sys

What is ln(x) ?

In short math words, ln(x)is the integral of the quadratic hyperbola 1/x i.e: the area under 1/x from the reference limit 1 ... ± b. To simplify your life, this area is computed via a numerical summing operator called "integrator". Several types of integrators are currently coded for you to use wisely. Mathcad integrator is TOL adjustable by user, others have default TOL not adjustable and of unknon accuracy.

x f x 1 b int b ln 21 mat 1.098612 1.098612 21 mat

Smath code ..... z Int 1 u / u 1 x int : x Σ z Int : 21 line

2 Σ 0.5 Σ 21 mat 0.693147 0.693147 - 21 mat

Can we see how this area sums over the evolution of 1/x ? Answer: YES, very easily => code the cumulative integration. From this nice data set LN, how to get a representative function ? Do nothing, mathematicians have done/named the function => ln(x). It resides in machine code in any computing machinery ... [Excel, any CAS, your pocket calculator ...].

Create/accumulate area LN x 0.01 5 0.05 range : i 1 x rows range A i el x i el Σ : for x A augment 31 line :

Cumulative area under the quadratic hyperbola 1/x. LN extends up to our limit, ln(x) extends as far as the plot canvas

LN x ln 1 5 1 5 - 2 2 mat 3 1 sys

<= axis of symmetry of the quadratic hyperbola

Revisit some interesting properties

x 1.23456789 : y 9.87654321 : 21 line

x y * ln exp y x / ln exp x y^ln exp 31 mat 12.193263 8 8.014042 31 mat

x y * y x / x y^31 mat 12.193263111264 8.0000000729 8.014042225511 31 mat

x ln y ln + exp y ln x ln - exp y x ln * exp 31 mat 12.193263 8 8.014042 31 mat

Something new in the ball game "exp(,)"

y exp x ln y - 0 ≡ x 0.01 e solve :

e 2.71828182845905

x ln x exp 1 e . 15 black 1 5 mat 3 1 sys

In this learning step, exp(x) is the InverseFunction of ln(x). An inverse function is a solving function. Generally a difficult task or not directly possible. Many common functions have their inverse built-in sin(x) <= asin(x), cos(x) <= acos(x) ... ln(x) <= exp(x). As specified in the solve block, it inverses ln(x) in the specified range 0.01..e where 'e' is the Euler constant e = 2.71828182845905. Note that in the plot, ln(x) keeps running over the plot canvas. The Smath "solve" is borrowed from Maxima, the new version of Macsyma. At some point during the design of the symbolic engine "Macsyma", the Toronto University went apart creating "Waterloo Maple", Maple from "MapleSoft" ranks world's first CAS, though so so easy for everyday Engineering works.

Conclusion:

All that said in less than two pages, we have learned about the two most fundamental "primary functions", doing simple everyday Engineering works as well as advanced projects. These two functions are built-in as they appear in what computes for you. Like other built-in functions, they crunch only the 4 arithmetic [+, -, *, /]. Commonly, they run on 21 decimals floating points of which only 15 are retained as "surely true". More floating points are available from ChebyShev polynomial expansion, which they exhibit higher stability at higher floating decimals. To satisfy your curiosity, ln(x), exp(x) each one crunches about 21 Aops [+, -, *, /]. Aops = "Aritmetic operations". The x^y is called "Power Function X^Y ", also a buit-in function as you see from above. So, this one crunches ~ 42 Aops. Typical, Chebyshev 25 floating decimals crunch ~ 75 Aops. Not sure how many "true" are retained.

A sense of smallness :

radius of Hydrogen atom 25*10^-12 m thickness of "one atom film" 50*10^-12 m the 15 decimals scientific caliper can easily determine of how many films a layer is made of.

Supplement:

All mathematical functions are approximations. Not a single one is "absolutely true", no matter zillions decimals you may collect. Most if not all are obtained from rational fractions [ratio of two polynomials]. Pn(x)/Qn(x) are numerically unstable at high order 'n' of the P, Q polynomials. The form exploited is a modified form "normalised rational fraction" Pn(x)/[1 + Qn(x)], more numerically stable. Let's satisfy our curiosity with Bessel, solution(s) of a class of DE in Physics.

JZERO 5852

|ε| = 23.22 decimals, [x] 0..8

p 1641556014884554375346147.435 394355966476729663601261.6471 - 21720183859245393139822.87997 481485995206981764828.5245941 - 5345457598841972345.381674607 33015389256896376.8646542622 - 118739068121104.2949874031474 247985116789.614443968987514 - 280314894.0831953934479400118 133662.5500481224741885945416 - 101 mat :

q 1641556014884554385346137.617 16033037244408932735390.45602 79130437776464052403.23616203 2613165313325.153278086066185 642960791882601.77592892131 1237672982083.40790348317773 1893012093.677918995179541438 2263381.356781110003609399116 1974.019272727281783930443513 1101 mat :

JONE 6054

|ε| = 25.02 decimals, [x] 0..8

a 1921176307760798128049021.316 222609203138739625477137.5773 - 7894463902082476734673.226741 126942437345360606543.6561036 - 1092152214043184787.101134641 5454629264396819.144157448868 - 16346594875712.84628830445048 29096627853.81647825756152444 - 28534334.51054763915026471449 11977.05712815379389149134705 - 101 mat :

b 3842352615521596256098041.912 35075670662720281057988.68716 161133431163341434400.7062889 492961231395985031.9632645381 1117536965288162.684489793105 1969278625584.719037168592923 2735606122.949877990248154504 2940957.355049651347475558106 2274.736606126590905134610965 1101 mat :

x J0 u x 2^11 line : p m el u m^* (m 110 sum q n el u n^* (n 110 sum / 21 line :

x J1 u x 2^11 line : x a m el u m^* (m 110 sum b n el u n^* (n 110 sum / * 21 line :

JONE can be safely extended to ~ twice its prescribed accuracy compared wrt Matlab "mwbesselj". This last one to be confirmed ?

|ε|= 23.22 decimals , [x] 0..8

x J0 x J1 2 1 sys

r x J1 x 1 - 18 solve :

rmw 1 x mwbesselj x 018 solve :

r 0 3.832 7.016 10.173 13.324 16.469 61 mat

rmw 0 3.832 7.016 10.173 13.324 16.471 61 mat