Euler’s Identity and the Eucalculus
By Charles Douglas Wehner |
4 July 2003 |
A learned English professor made the following statement:
In my analysis of this equation, which is known as the Euler Identity, I propose to show that it is true, yet all the other statements are false.
It was Colin McLaurin who first showed that an exponential may be represented
as a polynomial. He was closely followed by Brook Taylor, who gave a more
exhaustive analysis. The polynomial is therefore sometimes attributed to
either, but I propose to call it the McLaurin-Taylor equation:
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
It was Cardan (Latinised to Cardano) who first suggested the imaginary
numbers. Given a unit circle of X2+Y2=1, where is the
position X=1/2? Answer: Y is plus or minus ROOT (3/4).
And where is the position X=2? Answer: “Not on the circle”.
This did not satisfy Cardano. He wanted a mathematical answer,
and stated that Y would be at plus or minus IMAGINARY ROOT 3.
If 22+Y2=1, then Y2=-3. He had to
invent the square root of minus 1.
He defined all multiples of this imaginary root as imaginary numbers.
He decided that he did not need to invent a cube root, a fourth root,
or other roots of minus 1 because these could be defined in terms of
summations of existing REAL numbers and his new-found IMAGINARY ones.
Such summations, of the form A+iB - where A is REAL and B is IMAGINARY -
he defined as COMPLEX.
But -1 is one of the roots of 1. Thus, i1 - (imaginary 1) - is a FOURTH
root of one.
I propose to revisit the McLaurin-Taylor equation, and distribute it
into four boxes. This is my FOUR-BOX ALGORITHM.
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Where: Box1+Box2+Box3+Box4=eX These discoveries were made by McLaurin and Taylor, with the final one by Euler (pronounced “Oiler”). My contribution is simply a tidy FOUR-BOX presentation of these results. It can be seen that any power of FOUR times iX will be REAL (Box 1). Any power of FOUR, plus ONE, will be IMAGINARY (Box 2). Any power of FOUR, plus TWO, will be REAL NEGATIVE (Box 3). Any power of FOUR, plus THREE, will be IMAGINARY NEGATIVE (Box 4). Euler’s equation Cos(X)+iSin(X)=eiX is thus self-evident by inspection. We are left with the conundrum of how to visualise the equation in a memorable way. It is usual to present an X,Y graph as a REAL PLANE - such as on a sheet of paper. It has become a convention to add an IMAGINARY AXIS going INTO the paper. Negative imaginary is ABOVE the paper whilst positive imaginary is BELOW the paper. Examine this diagram: |
|
It can be seen that the argument X has been turned into a circle in the complex plane iY. This is seen as a red circle in perspective. The infinite number-line of the X axis (the domain) has been obliged to translate into a restricted co-domain with LOSS OF INFORMATION. The projection of this circle along the X-axis produces a UNIT SPIRAL in COMPLEX SPACE. The shadow of that spiral on the XY real plane is a COSINE. Note that the origin - at x=0 - is marked with an orange dot. The shadow of that spiral on the Xi complex plane is a SINE. Note that the origin (orange dot) is no longer at a crest of a wave, but in the middle of a wave, where SIN(0)=0. The diagram is marked “WEHNER” because there does not appear to be any prior art in explaining the Euler equation as the “EULER SPIRAL”. Returning to the special case of eiX, where X is Pi, we can see that it is at the BOTTOM of the spiral.
We can see the “PROBLEM OF MULTIPLE SOLUTIONS” when we ask, instead of
“What is the antilog of unreal Pi?”, which is -1, the exact OPPOSITE
question. “WHAT IS THE LOG OF -1”?
That is the same problem as saying “If the place on the spiral is -1,
where is X”?
ANSWERS: Pi, 3Pi, 5Pi, 7Pi... in fact ALL ODD MULTIPLES of Pi, AND THEIR
NEGATIVES.
So we can use the Euler identity to convert unreal Pi into -1,
but can only convert -1 back into unreal Pi if we know that is
where the -1 came from originally. This is a very limiting form
of mathematics!
There is also a COMPLEX form of the Euler equation, of
Y=eA+iB.
This relies on the equation
Y=eA+iB=eAeiB.
Pictorially, the spiral is no longer a unit spiral. Its magnitude
depends on the scaling factor eA.
So in the next diagram, the spiral is wrapped around an EXPONENTIAL HORN.
To avoid any confusion with perspective, the horn is exponentially
DECLINING (A is negative):
|
|
For any magnitude of horn (e to the A) there is a unique position X. Thus, we can say that for REAL numbers the logarithm is a one-to-one function. As B varies, however, the spiral (antilog iB) slides along the exponential horn. It takes up an identical position for every cycle of 2π. The logarithm is therefore a one-to-many function with regard to imaginary numbers, with consequent ambiguity. Other theorems, such as de Moivre, become much clearer when seen in this context. Professor Stephen Hawking is NOT the professor criticised in the opening paragraphs of this paper. In his book “A Brief History of Time”, he speaks of a curved universe. However, he even touches upon unreal time. Without some mathematical background, such as by reading this paper and learning that the terms UNREAL and IMAGINARY both refer to multiples of ROOT -1, the lay reader will be rather confused by the concept of unreal time. However, Charles Wehner had used the mathematics of i-omega-t in his electronics (where j-omega-t is used because i refers to current). The antilogarithm of j-omega-t is known to electronic engineers to give a cosine (SIGNAL) and an unreal sine (PHASE). One cycle of an oscillation in the time domain is very much like another, so the one-to-many ambiguity is no problem. However, if you are travelling in space along a distance line towards a star you cannot simply translate that distance-line to a time-line. All positions along that distance-line are not equivalent. Each is uniquely specified by its distance from that star. The Euler equation is therefore useful for REPETITIVE CYCLIC FUNCTIONS. It is also perilous to apply it to descriptions of the Universe. The curvature in Hawking’s curved Universe may be no more than the CURVATURE OF THE MATHEMATICS USED TO DESCRIBE IT. If Professor Hawking did indeed use “e to the i-omega-t”, he could just as well have spoken of REAL time and IMAGINARY ANGULAR FREQUENCY (i-omega). But it does not sound quite so romantic.
We began this story with the McLaurin-Taylor equation:
| ||||||||||||||||||||||||||||
which is a polynomial of the form: A factorial is made by multiplying numbers starting from 1. Thus, 4! will be 1 * 1 * 2 * 3 * 4 - that is, 24.
For example, if a function is exponential in one million, you will multiply by a million, and again by a million, and again by a million to get successive terms in the exponential series. To find factorial one million and one, you multiply factorial one million by the number one million and one. The next factorial will be that multiplied by one million and two. Although the factorial will be nowhere near the millionth exponential of one million, at least the rate of growth will be similar, because the successive multipliers only differ from each other by parts per million. Using this principle, Euler estimated that the factorial of a million and a HALF would be about a THOUSAND times the factorial of a million. He also decided that if you took a magic number for factorial A HALF, and multiplied it by 1.5, by 2.5 and so on until you reached 1,000,000.5 you should have that factorial of a million and a half. Accordingly, if you divide by 1,000,000.5 and then by 999,999.5 and then by 999,998.5 and so on until you come down to 0.5 you should find that magic number FACTORIAL A HALF. By deeper and deeper studies of this kind, Euler set out to create a CONTINUOUS form of the factorials, which he called the GAMMA FUNCTION. Working for hour after hour with poor eyesight, by candlelight, Euler SUCCEEDED - although ultimately he went blind. |
Euler took a particular interest in this function. He noted that when
a factorial becomes very large, it approximates to an exponential.
|
Euler found that the magic number was ROOT-PI DIVIDED BY 2, and that the factorial of MINUS A HALF was ROOT PI. Euler found that the numbers go crazy when you try to find the factorial of -1 or less. So he made -1 the origin of his curve (marked GAMMA 0 on the diagram). Three hundred years later, the author used powerful computer techniques to investigate the FRACTIONAL FACTORIALS, and also discovered root pi. He set the origin to 0!, however, for compatibility with the INTEGER factorial - as also marked on the diagram. The author created the standard of having a COMMA under the exclamation mark - as shown - to denote that this function allows the use of FRACTIONAL numbers. A standard exclamation mark signifies the usual FACTORIALS. The author gave the name FRACTORIALS to the function, showing that it combines both the FRACTIONS and the FACTORIALS. A particular use of the FRACTORIALS is in pocket calculators. Not only can you subtract 1 from the argument and so obtain the gamma function, but you can compute NEAR-INTEGERS without getting an ERROR-HALT. Current calculators test for an integer before permitting the factorial to be evaluated. This is fine if the user keys the number in. However, if it is the result of a computation there will be problems. Consider that you type in the number 1, and then DIVIDE by ten, and then MULTIPLY by ten. The display shows 1 again - but the factorial refuses to work.
This is because ONE TENTH is a RECURRING number in binary. Internally,
the calculator stores
for as many places as there is space. Multiplied by ten, you now get or similar. In DECIMAL, this is This is rounded to 1.0 simply FOR DISPLAY, but remains in the registers as a non-integer. The factorial program rejects this. Here is a QBASIC program, valid to about 12 places:
Change the GREATER THAN and LESS THAN symbols at UPDOWN from their
Hypertext equivalents if you want to run this program.
Note that from the curve given above, the factorial of -1 or below
could be either plus or minus infinity. The program simply says
“INFINITY”
If you follow the line “INPUT x#” by “x#=x#-1#”,
it becomes a GAMMA FUNCTION program.
Coefficient 2 in this form of the fractorial polynomial is the
SLOPE of the fractorials at zero, or of the gamma at 1. Euler gave it
the name LITTLE GAMMA. Its correct value is 0.577215664.
The author had computed little gamma with powerful programs to an
accuracy of 29 places of decimals, and could have found it to almost
any desired accuracy.
Working three hundred years ago, long before computers, and with
failing eyesight, Leonhard Euler calculated this number to SIXTEEN
places of decimals - a PRODIGIOUS achievement.
There exists the HARMONIC SERIES,
which are the ratios of the slope of the fractorials to the value
of the factorial at each integer step.
Thus harmonic nil is minus little gamma. Harmonic one is one minus
little gamma, or 0.422784336. Harmonic two is this plus a half, or
0.922784336. Harmonic three is this plus a third, or 1.256117669.
The author created high quality coefficients for this also,
and so produced the continuous
HARMONIC FUNCTION.
Note that the above coefficients are CHEBYSHEV (TSCHEBYSHEV)
coefficients, as described by Panyut Chebyshev, Professor of
Mathematics at St. Petersburg, Russia in the 19th century.
These numbers are “warped” to allow a shortened polynomial
to behave over a restricted range like an infinite one.
Returning to the diagram, and studying the red
lines, one sees that at -1 PLUS “ZERO” (a trifle), the fractorial is
INFINITE whilst at -1 MINUS “ZERO” it is MINUS INFINITE. One could even
describe the slope of the red lines as INFINITE SQUARED, because they
jump by twice infinite for a change in X of twice zero.
This non-standard terminology comes from a further study made by
the author, where he discovered the transfinite numbers. The route
was a different one from that taken by Georg Cantor in the 19th
century when he discovered them first. However, it shows that
transfinite numbers must be in nature, or they could not be
independantly discovered twice.
In the author’s FANTASY MATHS, zeroes and infinities are defined as
“FOLLIES” because they can make the maths go awry. However, other
numbers are described as “FUN”. Combinations of “FUN” and “FOLLY”
are described as “FANTASY”. For example, the McLaurin-Taylor
equation is a fantasy equation because although the individual
terms are “FUN”, in that they may be worked out, there are an
INFINITE number of them.
It is a fantasy to dream of evaluating a series until the end of
time. However, a truncated McLaurin-Taylor series can be fun.
The Chebyshev polynomial for the natural antilogarithm may save
ten percent or so when computed in a calculator’s floating-point
system, but if maximum speed is not essential, the judicious use
of a truncated McLaurin-Taylor series will serve because the
coefficients are memorable as the reciprocals of factorials.
The author created polynomials for the reciprocals also.
He filled a book with Chebyshev coefficients for both
the fractorials and their reciprocals, optimised for each
degree of accuracy from four decimal places to twenty-four.
Here is the reciprocal graph:
|
|
The crazy swooping curves of the gamma function have been tamed
into what look like sinusoids. The red lines - plus infinity
above and minus infinity below - have each become a
dot as the sinusoids sweep smoothly
across the x-axis.
This curve is the LOCUS of the AMPLITUDE of a COEFFICIENT during INTEGRATION, with integration represented as a movement to the RIGHT. For example, if we have Y=1 at the point X=0, we can scale this by 1 because that it the reciprocal of factorial zero. After HALF an integration, we arrive at the point X=½ where the eucalculus function is 2 divided by root π. That is now the size of the coefficient. After a further HALF integration, we arrive at X=1. Here, the factorial of 1 is 1, so the coefficient shrinks back to what it started as. Of course, exactly the same thing happens in the reverse direction during differentiation. GAIN is provided by the hump. Consider the following electronic novelty circuit: |
|
One might set the value of R
to the parallel combination of R1
and R2. This reduces the analysis
to that of six identical CR
differentiators fed from a low source impedance
R3.
The output of those differentiators goes to the base b of the transistor VT1 to emerge at higher current at emitter e. There is no voltage gain in this “EMITTER-FOLLOWER” style of circuit, so there should be a “LOOP-GAIN” just below unity, and the circuit should not oscillate. Yet designs of this type have been made to work. Four differentiations are required - from SINE to COS, from COS to -SINE, from -SINE to -COS and finally from -COS back to SINE. Each would be a FULL differentiation, and would spring over the “HUMP”. So four or less CR stages cannot deliver oscillation. However, with more than four stages the system must find some frequency at which the loop phase is 360°. Four differentiation delivered by six differentiators implies that each stage gives only TWO THIRDS of a differentiation, with a phase-shift of 60° per stage. Under these conditions there will be some voltage gain available. Whereas the Hartley oscillators can be explained by the transformer action of their inductors, the Colpitts family of designs often seem to deliver gain from capacitors as if by magic. Although the analysis has been far from rigorous, it has at least provided the flavour of the new “Eucalculus” with its study of FRACTIONAL integration and differentiation. The prefix Eu is Greek (pronounced Ef in Greek as in Efkaristo for “Thank you”, but pronounced Yoo in English, as in Eucharist, the “Thanksgiving ceremony”. The word Calculus is Latin. Nevertheless, the author has combined the two words to represent the GOOD COMPUTATIONS, a SUPERSET of the conventional calculus, because it combines the concepts of the traditional calculus and the new fractional variants. The author had studied the “HUMP” to great precision, to evaluate the highest gain obtainable by fractional differentiation - but no longer has access to his records. The ZERO CROSSINGS on the Eucalculus curve are interesting, as it is into these “holes” that the constants disappear during unit differentiation.
For example, X cubed differentiates to 3 times X-squared, Wherever one of these factors is zero, it CONCEALS all the other factors that accompany it. In the REALMS of the Fantasy Maths, and in the ALEPHS of Cantor, the accompanying factors still exist - but the function has been reduced to zero. Those factors would only reappear if that special zero is divided by a zero IDENTICAL IN SIZE TO THE ZERO FACTOR THAT CAUSED THE PROBLEM. This is totally non-standard mathematics. Thus, without reference to special zeroes, the problem of the INTEGRATION CONSTANT remains. A function, once differentiated, loses information that cannot be recovered by integration. However, on either side of these red dots there are curves which the author, on first seeing them, described as “PEEP-PO” curves. It is as if the Integration Constant was a small child hiding behind a tree. Children often play “PEEP-PO” to the left, and then to the right. As they get more excited, the amount by which they extend themselves out of hiding increases. So it is with these apparent sinusoids, the further you journey to the left. Are the “PEEP-PO” curves truly sinusoids? The analysis continues with the concept of evaluating a factorial. Factorial 2 can be raised to factorial 3 by multiplying by 2+1. It can be reduced to factorial 1 by dividing by 2+0. Or in the case of the Eucalculus function, we go up to 3 by DIVIDING by 2+1 or go to 1 by MULTIPLYING by 2+0. It follows that 2+½ is a special number. Fractorial 3½ can be obtained by multiplying by 3½, and fractorial 1½ by dividing by 2½ - but we see nothing special yet. Perhaps 1+½ is a special number. We go up with 2+½ and down with 1+½. Nothing yet. What about ½? We go up with 1+½ and down with ½. What about -½? We go up with ½ and down with -½. THAT’S IT!. There is INVERSE SCALING SYMMETRY about the point -½. Those “PEEP-PO” curves are growing FRACTORIALLY as we travel to the left. Similarly, the smooth curve over the hump is declining FRACTORIALLY as we travel to the right. The author multiplied the left side of the Eucalculus curve by the right side, symmetrically about -½. The result was a COSINE. But what is this cosine? Is it a SHADOW ON THE WALL such as we saw with Euler’s spiral? It is COS πX divided by π. With all trace of fractorial growth and shrinkage removed, we need only explain the ROOT π and the COS πX. James Stirling also discovered the ROOT π. It occurred to the author that the correct approach would be not to investigate the fractional factorials, but the fractional factorials SQUARED. This would give an amplitude of π at -1.
Here are some coefficients for the fractorial squared, with -0.5
(or +0.5 on the Gamma graph) as origin: |
3.141592653588666 -12.31952125062089 39.17187545092207 -102.7591818905219 225.7860640502422 -396.5945759097769 528.08152858418 -503.7989073627172 320.9902434783365 -121.4629532270357 20.54923358680391 |
|
Bear in mind that the expressions “b# = b# * a#” and “b# = b# / a#” will have to be doubled up if you substitute these coefficients in the Qbasic program. The coefficients deliver π at zero, so the first is π. They deliver π by four at 1, so they summate to π/4. There are errors due to limitations of the generator program. The quest is for SIMPLE explanations. For example, π divided by 2e is 0.577863674 approximately. It looks like little gamma, except that after the 0.577 the two numbers diverge from each other. When such numbers appear, one asks whether they are π over 2e WARPED by the Chebyshev approximation, little gamma WARPED by the Chebyshev approximation, or something else. To this end, the author created a 32-decimal-place mathematical system, complete with sinusoids, arcsinusoids and logs, and another with 65 places of accuracy - as laboratory standards. Pythagoras discovered that a right-angle could be made by means of a 3-4-5 or a 7-24-25 or any of a series of “Pythagorean” triangles. His disciples were sworn to secrecy, so that they could go out at night to building-sites and create a right-angle for the now legendary Greek architecture. Had anybody broken the code of secrecy, the triangles would be known today as “builders’ triangles”, Pythagoras and his followers having starved to death for lack of payment.
Pythagoras, whilst passing a blacksmith’s shop noticed that some
sounds were sweet, whilst others were hideous. Investigating
“harmony”, he discovered that an The resulting HARMONIC SCALE survived for thousands of years until the arrival of Johann Sebastian Bach, who introduced the EQUITEMPERED SCALE. It seemed as if the very SOUL of Man, in its appreciation of music, was based upon the harmony of simple numbers. Socrates, Plato and Aristotle debated the question of whether the soul is a harmony. If the harp is broken, whence goes the harmony? According to Aristotle, PERCEPTION ITSELF is a ratio. If it is not a ratio, it is IRRATIONAL. So when a disciple of Pythagoras insisted that there were irrational numbers, Pythagoras grew afraid that they would stop making discoveries and lose their livelihood. He ordered that the man be put to death. We know today that there are indeed irrational numbers that go on forever, with NO REPEATS. The base of the natural logarithm, e, is just such a number. Here are just the first 1000 figures: |
2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945 713821785251664274274663919320030599218174135966290435729003342952605956307381323286 279434907632338298807531952510190115738341879307021540891499348841675092447614606680 822648001684774118537423454424371075390777449920695517027618386062613313845830007520 449338265602976067371132007093287091274437470472306969772093101416928368190255151086 574637721112523897844250569536967707854499699679468644549059879316368892300987931277 361782154249992295763514822082698951936680331825288693984964651058209392398294887933 203625094431173012381970684161403970198376793206832823764648042953118023287825098194 558153017567173613320698112509961818815930416903515988885193458072738667385894228792 284998920868058257492796104841984443634632449684875602336248270419786232090021609902 353043699418491463140934317381436405462531520961836908887070167683964243781405927145 63549061303107208510383750510115747704171898610687396965521267154688957035035 |
|
To find e, you take 1 and put it in the potThere is always a unique new divisor, so there is always a new way that the number is being extended. It is a branch of CHAOS THEORY in that although the algorithm is simple, the scale of chaotic variation that results quickly becomes impenetrable to the human mind. The value of e was actually generated by Horner’s Algorithm. This was discovered by Isaac Newton and myself, but Horner published it as a feature. You put 1 into the pot and divide the pot by 1000. Add 1 to the pot, and divide the pot by 999. Keep going (backwards). You can see this in action in POLYNOM, the polynomial evaluator, in the Basic program. The search was for a Pythagorean solution to the fractorial problem - some equation like the McLaurin-Taylor formula for the antilogarithm, in which the coefficients are Diophantine multipliers or Diophantine divisors - whole numbers. If discovered, this would reveal a law of nature. The presence of root pi at -0.5 certainly suggests that something profound is waiting to be revealed. There was no time to investigate further the origin of the root pi, or to seek out a simple route to the fractional factorials such as via their squares. Tragedy was soon to be inflicted upon the author. The author attended to that other facet of the fractorials. Above -0.5 they are always seemingly multiplied by a positive number, but below that point each step in descending by unit distance involves a negative multiplier. If we have a PROCESS, such as exponentiation in 4 - giving 1, 4, 16, 64 &c. - there should always be a SUB-PROCESS, such as a HALF-PROCESS. The half-process of exponentiation in 4 is exponentiation in 2. This gives the sequence 1, 2, 4, 8, 16, 32, 64 &c. which begins to fill in the gaps. The half-process of exponentiation is exponentiation by the ROOT of the base. Thus, the gaps in the sequence 1, e, e-squared can be filled in by powers of ROOT e. The one-third process of exponentiation involves the CUBE root, and so on. The investigation of the fractional factorials is therefore a study of the subdivision of the factorial process. We have discovered an exponentiation in -1, which the author calls the negation function. It gives 1, -1, 1, -1 &c. - and its half- process is therefore exponentiation in ROOT -1. That is to say, we exponentiate in i1. We want the function -1X for all values of X. It is customary to take the LOGARITHM of the number, multiply by X and find the ANTILOGARITHM. However, Log-1 to base e is π, 3π 5π etc. and all negative values.
Taking π and multiplying by X gives us πX. Exponentiating by
Euler’s equation gives Cos(πX) + iSin(πX). This is just like
Euler’s spiral - but π times as dense. It is also the frequency
of the PEEP-PO curves. |
|
Nature certainly seems to have taken one of the fundamental solutions to the problem of multiple solutions. Nature chose a single π - but is it a positive, or a negative? Only the discovery of the SHADOW ON THE GROUND, of the sine or the minus-sine, will settle the question as to whether the Eucalculus function spirals in complex space. If it does, then the fractorials - the gamma function - must also spiral in complex space.
The reciprocal of a unit spiral in complex space is an identical
spiral of opposite hand: |
|
If the Eucalculus function begins to spiral at -0.5, there are two questions: HOW DOES IT BEGIN WITHOUT A KINK? and IN WHAT HAND?
Nature abhors a kink - a SINGULARITY.
The presence of such a kink casts doubt upon the validity of the
function as a whole: |
|
Yet, despite the most extensive study no indication could possibly be found that there is an imaginary sinusoid within the Eucalculus formula. It is as if Nature refuses to imagine. A beginning was made to study the HALF fractorials - the process that gives the half-way step between the factorials. We have this already, but it is not just a set of coefficients that is required but a deeper UNDERSTANDING of the process involved. If the half-process involved a complex function of the argument it could be stated that spiralling begins at zero and continues for all negative values. The inverse scaling symmetry, however, suggests that spiralling takes place at -0.5 and below. The kink would show up in the iX complex plane. The “SHADOW ON THE WALL” of the Eucalculus function would be identical to the XY graph shown previously.
If there is such spiralling, the reciprocal of the Eucalculus
function - and by implication, the Gamma function, must also spiral: |
|
The hand of the spiral is opposite to that of the Eucalculus function. The kink would be visible in the iX complex plane. The image in the XY real plane is identical to that of the fractorial (gamma) function for values above zero (fractorial), which is 1 for gamma. Below this point, where spiralling has become established, there is no longer any resemblance to the established gamma curve. The junction between the smooth right-hand curve and the spiral left-hand curve is marked by a kink even in the XY plane. To avoid this, neither -0.5 nor 0 can be the spiral-onset point on the fractorial X-axis. We can see that there is a choice either to believe in the transfinite numbers, which produced the red lines reaching from plus infinity to minus infinity, or to believe in complex mathematics. 1 divided by NIL is infinity - but if there is an invisible imaginary component i1, we must compute 1 over 0+i1.
Such a computation delivers -i1. This is far removed from infinity.
The solution to this paradox is important if the gamma function and its attendant functions - such as the Bessel series, which have proven so useful in astronomy - are to have a firm foundation. The riddle also casts further doubt upon the complex mathematics, and upon the new Eucalculus which is now introduced. Issac Newton’s FLUXIONS and Brook Taylor’s CALCULUS OF FINITE STEP appear to be virtually identical. During the Great Plague of 1665, Newton shut himself away. He seems to have toyed with a fob-watch to pass the time, and found himself counting the swings. Eventually, during that year, he was using ever longer pendula and building up sizeable tables that he analysed.
I shall not attempt exactly to replicate those tables,
but shall take hypothetical examples and show them in reverse:
...where times are given in seconds. Taking the length
column on the right, you subtract each entry from the one below and put
the result in the column to its left. You subtract these figures and
transfer them also to the left.
Thus, a subtraction implies a movement to the left.
Newton obviously noticed a similarity to the function X-SQUARED:
Yes! after just two columns of subtraction you have a constant - in this case FACTORIAL 2. So Gravitational acceleration is a second-order function. It involves time to the power of TWO. Each subtraction takes the function down one order. When you reach the zero order, you can estimate the coefficient. However, a single x-squared delivers 2!, so you have to divide the zero-order difference by FACTORIAL 2. That is how Newton found A HALf f-t-squared. You must also divide by the step-size for each order. Here it was one tenth, so the final constant times one hundred, divided by the factorial of 2 gives the acceleration due to gravity in inches per second per second. You can use third, fourth, fifth or any order of equation. The constant that arises is divided by the factorial of the order to provide the coefficient.
Consider this table:
It is a third-order truncated form of our old friend, the
McLaurin-Taylor equation:
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
The bottom entry on the left resolves to 1. So we divide by the
factorial of 3 (by 6), and obtain the third-order coefficient.
Components of the numbers have been kept separate. So we can
simply strike out all third-order components on the right and
look at the diagram again.
The second column from the left, with third-order parts removed,
resolves to two second-order entries each of which simplifies to 1.
Divided by 2 these give the second-order coefficient.
Now we can strike out the second-order parts and
carry out the differencing of a single column to obtain the
coefficient of X, which is 1.
With that removed, we have only the constant -
which is also 1.
But supposing we did not have four pieces of data in the right-most
column, but only three. We would be forced to treat the function as
a SECOND-order one instead of a third.
Our final result would be:
This resolves to 3, not to 1. So we would be feeding back three times too much of the second-order component, making the data too small. The result is that the first-order coefficient would be too small. When the first-order coefficient is fed back its smallness helps reduce the data-error caused by the excessive second-order coefficient. Then we can do no more subtractions, so what error there is remains.
The result would be a SECOND-ORDER
CHEBYSHEV APPROXIMATION TO A THIRD-ORDER POYNOMIAL.
It would be valid only within the data-set used to create it. That is to say, the range 1-2-3. Try to evaluate a 0 or a 4 and the numbers go crazy. Not all functions resolve to a Chebyshev poynomial. They do if they are WELL-CONDITIONED, but in an ILL-CONDITIONED system the error, when fed back, gets worse. Here are some Chebyshev polynomials for a cosine. Note that they should be 1, 0, 0.5, 0, 0.0416666666666666 &c., but the Chebyshev feedback has thrown the error into the zero coefficients which in turn has reduced the error in each preceding coefficient.
These coefficients are valid over the range zero to 1.
It is possible to do TWO subtractions of the tables, thereby springing over those coefficients that should be kept as zero. The coefficients appear to be worse, but with a specialised form of the Horner algorithm can deliver a faster result with no loss of quality.
In that form, you generate X2 first. Then you multiply
-2.4087D-07 by X2. You add 2.4574D-05 and multiply by
X2, and so on until you add the 1 and stop. It takes
the initial multiplication plus five - instead of ten multiplications.
The way in which the non-zero coefficients 1, -0.5, .0416666 &c. alternate between plus and minus in a cyclic function like a sinusoid was first spotted by Colin McLaurin. Charles Babbage noted that if you set up a table of differences, you can add the numbers MECHANICALLY as long as they are based upon natural laws rather than Chebyshev style approximations driven beyond their allotted domain.
Here is the square-law table again:
In such a case, a Babbage DIFFERENCE ENGINE would add the final difference (2) to the next difference (9) to obtain 11. This, when added to 25 gives 36 - the next square. Then 2 plus 11 plus 36 give 49. Babbage realised that such functions as the sine and cosine were of great importance to the Admiralty for navigation. When type-set by hand, one in five numbers was wrong. However, a machine that printed as it computed - with no human intervention - would deliver perfect tables to any accuracy desired. Unfortunately, Babbage hired a mechanic who belonged to the “Disgusting Gang”, who had previously cruelly exploited Isambard Kingdom Brunel. This criminal milked Babbage of money, and did no work. Britain never made the world’s first computer. The reader will have discovered by now how profoundly productive the world of subtraction-tables has been.
In the Calculus of finite step, we had
DY/DX
for the ratio of a difference in the data to the size of the step.
In the Infinitesimal Calculus, we had
Note that neither
Process P is indeed a number. It specifies
how much integration is being performed. 1 signifies a single full
integration, whilst 0.5 signifies an integration process which -
if repeated - would give a single integration.
It is also a vector quantity, in that -1 signifies a single full
differentiation. Microfine subdivisions of the differentiating
process are possible as with the integration process.
In this way, integration and differentiation are unified.
It was considered that as process P
is not a multiplier, it should not be on the same level as
X. It should be a
SUPERFIX.
However, although it is P in the general
formula, it becomes a number in any specific case. A further symbol
is required to signify what kind of process is employed.
This symbol is a SUFFIX. The use of a Greek
Sigma or other special symbol might put too much strain upon the
typographer’s art - already burdened by the special exclamation
mark.
The decision was taken to use a small m,
signifying microfine integration,
micro integration or
mini integration.
Here is the general formula: |
|
It will be observed that when a single integration is required,
the process P is unity: |
|
Here, q! divided by (1+q)! resolves to 1/((1+q)!). This is identical to the result for standard integration. You would seek out position q on the X-axis of the Eucalculus curve, where the Y value represents the magnitude of Xq. Slide one unit to the right, and read off the value to find the magnitude of X1+q.
When the process P is -1, we have a
standard differentiation: |
|
Here, q! divided by (q-1)! resolves to q. The result is exactly the same as for standard differeniation. You would seek out position q on the X-axis of the Eucalculus curve, where the Y value represents the magnitude of Xq. Slide one unit to the left, and read off the value to find the magnitude of Xq-1. Notice that the dX-1 of the Leibnitz infinitesimal calculus is preserved in the Eucalculus. With the Gamma Riddle unresolved, those who wish to explore these new concepts are referred to the Qbasic program for the transfinite, or non-spiral, form of the fractional factorials.
Functions such as the sine and cosine can be treated as their polynomials.
However, there is a BILATERAL form of every
polynomial of this kind, such as of McLaurin-Taylor:
| ||||||||||||||||||||||||||||||
Let us examine the right-hand side of this series:
| ||||||||||||||||||||||||||||||
The top is exponential in X. The bottom is factorial. Factorials are
HYPERexponential in that they have a
RISING BASE, whilst the base of an
exponential is fixed.
Whatever the value of X, in a fantasy series like this the factorial
will ultimately have risen to exceed it. From then on, the successive
terms become smaller and smaller as the series
CONVERGES.
It is a different situation with the left-hand series, however:
| ||||||||||||||||||||||||||||||
Here we have nominators of X to the -3, -2 and -1 which are really
denominators of X to the 3, 2 and 1.
Similarly, reverse scaling symmetry about -0.5 gives us this equation:
|
|
If we take 0.5 away from each value of X, or add 0.5 to -X, we have a new simplified equation: |
|
So it can be seen that the factorials of negative numbers used as denominators are better understood as the factorials of positive numbers, scaled by pi, modulated by sin pi.X, and used as nominators.
The nominators and denominators therefore have been swapped, as
compared with the right-hand side.
| ||||||||||||
For integer values of X, Sin(πX) is nil. The series will resolve
to zero.
If X is a non-integer, however, the sines will play “PEEP-PO”.
But more than that .... with the top f(r)actorial and the bottom
exponential, the series will DIVERGE towards the left.
Now there’s a thing........!
|
 |
| HOME |
(C) 1985-2003 Charles Douglas Wehner.
Use freely but do not plagiarise.