1 - e-x
Some Ramana Maharshi quotes:
The nature of bondage
is merely the rising, ruinous thought "I am different from the Reality."
Since one surely cannot remain separate
from the Reality, reject that thought whenever it rises.
You speak of memory
& oblivion of the fullness of the Self. Oblivion & memory are only thought-forms.
They will alternate so long as there are thoughts. But Reality lies beyond
these. Memory or oblivion must be dependent on something. That something must
be foreign to the Self as well, otherwise there would not be oblivion. That
upon which memory & oblivion depend is the idea of the individual self.
When one looks for it,
this individual "I" is not found because it is not real. Hence this "I" is synonymous with
Illusion or Ignorance [maya_, , or ajnana].
To know that there
never was Ignorance is the goal of all the spiritual teachings. Ignorance must
be of one who is aware. Awareness is jnana. Jnana is eternal & natural, ajnana is
unnatural & unreal.
In Deep Sleep man is
devoid of possessions, including his own Body.
Instead of being
unhappy he is quite happy. Everyone desires to sleep soundly.
The conclusion is that
Happiness is inherent in man & is not due to external causes.
One must realize the
Self in order to open the store of unalloyed Happiness.
lf a man thinks that
his Happiness is due to external causes & his possessions, it is reasonable
to conclude that his Happiness must increase with the increase of possessions &
diminish in proportion to their diminution. Therefore if he is devoid of
possessions, his Happiness should be nil.
What is the real experience of man ?
Does it conform to this view ?
Calculus for Yogis,
part 5
More interesting is a Series representation of the
Exponential function with e. Factorial notation helps to reveal the wondrous
pattern in the Series representation of e
x
:
∞
e x =
Σ x n / n! = 1 + x + x 2
/ 2 + x 3
/ 6 + x 4
/ 24 …. small x
n = 0 = x 0 / 0! + x 1 / 1!
+ x 2 / 2 x 1 + x 3
/ 3 x 2 x 1
+ x 4
/4 x 3 x 2
x 1
Such a Series allows easily estimating e
= 2.718281828
… that Transcendental
number.
Having represented the Exponential function in this series
sort of Polynomial form, we can, using the Power Rule as above, readily take
Derivative of each term in turn:
d y / d x of: y = 1, =
0
y = x, =
1
y = x 2
/ 2, =
x
y = x 3 / 6, = x 2 / 2
y = x 4 / 24, = x 3 / 6
& term for term,
we get back each prior term again. For
an infinite series, being a step behind won't matter. In all then, d y / d x of: e x = e x .
So this Transcendental number e is a "natural" base for the Exponential function just
because the Derivative, the rate of increase, goes up exactly with the value of
the function. So this Transcendental
number e is a
"natural" base for the
Exponential function since it goes up exactly with function itself, in the
sense of "the more you got, the more you get." Aside from a constant rate of increase, this
"more you got, the more you get" rate of increase is one of the
simplest & most "natural".
This also offers an entry point for looking at that "inverse"
of the Derivative, the "Integral", by first guessing that it must
"work both ways" for the Exponential function. If Derivative there gives us the Exponential
function back again, then vice versa
must also work, so the Exponential function is also its own Integral as
well as its own Derivative; & also we can take the term by term process
above & reverse each of those for Polynomial terms.
If d y /
d x
of: y = 1, =
0, then the indefinite Integral of Zero is a Constant, so then a
Constant can be added to every indefinite Integral since Zero is just
"nothing". Next we see that
Integral of 1
is x; that the Integral of x is x 2
/ 2; that the Integral of x is x 2
/ 2; that the Integral of x 2 / 2 is x 3 / 6; that the Integral of x 3 / 6 is x 4 / 24. Thus the
"inverse" of the Derivative for Polynomial terms, the Integral is
taken by raising the Power & dividing by that new Power [in contrast to multiplying by the original
Power & lowering that Power.]
In considering the "inverse" of the Derivative,
namely the "Integral", we happened upon some inverse relationships for simple Polynomial terms & their
components, such as: Division being the inverse
of Multiplying; Subtracting being the inverse
of Adding; raising a Power being the inverse of lowering a Power.
Well the corresponding though more complicated inverse of the Exponential function was
historically called the "Logarithm".
In shorthand, Log x is the inverse of 10x while inverse
of the Exponential function was historically called the
"Logarithm". In shorthand, Log
x is the inverse of 10x the inverse
ex
was dubbed ln x, with the "n" for "natural". Some simple Series result for ln x, such as:
∞
ln(1 – x) =
– Σ x n / n = – x – x 2
/ 2 – x 3
/ 3 – x 4 / 4 …. for small x
n = 0
Furthermore, the Exponential function for
"complex" numbers [to be
mentioned later] combines the simplest Trigonometric ["wave"] functions Sin x & Cos x so that series representation for each use
alternating odd & even
terms from the Series representation of the Exponential function. Thus:
∞
sin x =
Σ ( –1)n x 2n+1 / (2n+1) ! = x
– x 3 / 3 ! + x 5 / 5 !
– x 7 / 7 ! ...
n = 0 = x –
x
3 / 6
+ x 5 /120 – x 7 / 5040
...
∞
cos x =
Σ ( –1)n
x 2n / 2n ! = 1 – x 2 / 2 ! + x 4 / 4 ! – x 6 / 6 !
….
n = 0 = 1 – x 2
/ 2 + x 4
/ 24 – x 6 / 720 …
Taking the Derivative of each term in turn can show that:
d y / d x of: y = sin x , = cos x
d y / d x of: y = cos x , = – sin
x
The corresponding indefinite Integrals, respectively
are: – cos x & sin
x
This can again be show by taking the Integral of each term.
The above themes & 1600 pages more are freely available as perused or downloaded PDF’s, the sole occupants of a Public Microsoft Skydrive “Public Folder” accessible through:
or with Caps-sensitive:
Duplicates have been available on:
jstiga.wordpress.com/
[But from now on, they will be different & still usually daily.]
"There is no Creation, no Destruction, no Bondage, no longing to be freed from Bondage, no striving for Liberation, nor anyone who has attained Liberation. Know that this to be Ultimate Truth." – the "no creation"school of Gaudapada Nome – Ajata Vada
for very succinct summary of the teaching & practice, see: www.ajatavada.com/
"There is no Creation, no Destruction, no Bondage, no longing to be freed from Bondage, no striving for Liberation, nor anyone who has attained Liberation. Know that this to be Ultimate Truth." – the "no creation"
for very succinct summary of the teaching & practice, see: www.ajatavada.com/
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