Monthly Archives: September 2017

The Fantastic Fire Bird

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The fantastic fire bird fractal above is produced by Omega CAS Explorer:

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Introducing Lady L

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 Fig. 1

In my previous blog “A Case of Pre-FTC Definite Integration“, we obtained result

\int\limits_{a}^{b} {x^p} dx= {{b^{p+1} - a^{p+1}} \over {p+1}}, p\in Z_{0}^{+}\quad\quad\quad(1)

without the Fundamental Theorem of Calculus.

Let’s now consider the case of \int\limits_{a}^{b} {x^p} dx where p=-1.  Namely, \int\limits_{1}^{x}{1 \over u} du, the area under the curve from 1 to x.

Closed form result (1) is not applicable since p = -1 \in Z^{-}.

Attempt of finding the limit of a sum quickly bites the dust too due to the fact that \sum\limits_{i=1}^{n}{{x-1} \over n}{1 \over {1+{i{{x-1} \over n}}}}=\sum\limits_{i=1}^{n}{{x-1} \over {n + i(x-1)}}.

However,  let

\log{x}  = \int\limits_{1}^{x}{1 \over u} du\quad\quad\quad(2)

We see immediately that \log{1} = \int\limits_{1}^{1}{1 \over u} du, i.e.,

\log{1} = 0\quad\quad\quad(3-0)

Other properties of function \log{x} can be extracted from (2), as shown below:

By definition,

\log x_{1}x_{2} = \int\limits_{1}^{x_{1}x_{2}} { 1 \over s} ds = \int\limits_{1}^{x_{1}}{1 \over s} ds + \int\limits_{x_{1}}^{x_{1}x_{2}}{1 \over s }ds,

\log{x_{1}} = \int\limits_{1}^{x_{1}}{1 \over s} ds

and

\int\limits_{x_{1}}^{x_{1}x_{2}}{1 \over s} ds= \lim\limits_{n \to \infty}\sum\limits_{i=1}^{n}{1\over{x_{1}+i{{x_{1}x_{2}-x_{1}} \over n}}}{{x_{1}x_{2}-x_{1}} \over n}

 = \lim\limits_{n \to \infty}\sum\limits_{i=1}^{n}{1\over{ ({1+i{{x_{2}-1} \over n}}) x_{1} }} {{x_{1}(x_{2}-1)} \over n}

 = \lim\limits_{n \to \infty}\sum\limits_{i=1}^{n}{1\over{1+i{{x_{2}-1} \over n}}}{{x_{2}-1} \over n}=\int\limits_{1}^{x_{2}}{1 \over s}ds =\log{x_{2}} .

Therefore,

\log{x_{1}x_{2}} = \log{x_{1}} + \log{x_{2}}\quad\quad\quad(3-1)

Let x_{2}={1 \over x_{1}}, we have

\log{x_{1}{1 \over x_{1}}}=\log{1} = 0 = \log{x_{1}} + \log{1 \over x_{1}},

i.e.,

\log{1 \over x_{1}} = - \log{x_{1}}\quad\quad\quad(3-2)

By (3-2),

\log {x_{1} \over x_{2}} =\log{x_{1}{1 \over x_{2}}}= \log{x_{1}} + \log{1 \over x_{2}}

= \log{x_{1}} - \log{x_{2}},

i.e.,

\log{x_{1} \over x_{2}} = \log{x_{1}} - \log{x_{2}}\quad\quad\quad\quad(3-3)

Let p = 0,

\log{x^{p}}=\log{x^{0}} = \log{1} = 0 = 0\cdot \log{x} = p\log{x}\quad(3-4)

When p = 1,

\log{x^{p}}=\log{x} = 1\cdot \log{x}=p\log{x}.

Assume when p=k , \log{x^{k}}=k\log{x}  where  k \in Z^{+},  we have

\log{x^{k+1}}=\log{x^{k} x} =\log{x^{k}} + \log{x}

=k \log{x} + \log{x} = (k+1)\log{x}

Hence,

\log{x^{p}} = p \log{x}, \forall p\in Z^{+}\quad\quad\quad\quad(3-5)

Moreover, \forall p \in Z^{-},  p = -k where k \in Z^{+},

\log{x^p} =\log{x^{k}} = \log{1 \over x^{k}} = \log{1}-\log{x^k}

=-k\log{x} = p \log{x}

As result,

\log{x^p} = p\log{x}, \forall p \in Z^{-}\quad\quad\quad\quad(3-6)

With (3-4), (3-5) and (3-6), we conclude that

\log{x^p} = p\log{x}, \forall p \in Z\quad\quad\quad\quad(3-7)

We will leave this post with the following observation:

\lim\limits_{ h\to 0}{{\log(x+h)-\log{x}} \over {h}}={1 \over x}\quad\quad\quad(4)

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Fig. 2

This is not difficult to see. \forall x > 0, if h > 0, then from Fig. 2, we have

area in blue =\int\limits_{x}^{x+h}{1 \over u} du = \int\limits_{1}^{x+h}{1 \over u} du -\int\limits_{1}^{x}{1 \over u}du.

The fact that

{1 \over {x +h}} \cdot {h} < area in blue < {1 \over {x}} \cdot {h}

means

{1 \over x + {h}} { h} < \int\limits_{1}^{x+h}{1 \over u}du -\int\limits_{1}^{x}{1 \over u}du < {1 \over x} {h},

or,

{1 \over { x + h}} <{{\int\limits_{1}^{x+h}{1 \over u}du -\int\limits_{1}^{x}{1 \over u}du} \over {h}} < {1 \over {x}},

Since \lim\limits_{h \to 0} {1 \over {x +h}} = {1 \over x}, \lim\limits_{h \to 0} {1 \over {x}} = {1 \over x},

\lim\limits_{h \to 0}{{\int\limits_{1}^{x+h}{1 \over u}du -\int\limits_{1}^{x}{1 \over u}du} \over {h}} = {1 \over x}

from which (4) is obtained.

When h < 0, area in blue is

\int\limits_{x + h}^{x}{1 \over u} du = -\int\limits_{x}^{x +h}{1 \over u} du = -(\int\limits_{1}^{x + h}{1 \over u}du -\int\limits_{1}^{x}{1 \over u} du).

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Fig. 3

From Fig. 3 we see that

{-{h} \over x } <-({\int\limits_{1}^{x+h}{1 \over u}du -\int\limits_{1}^{x}{1 \over u}du})< {{-h}\over { x + h}}.

Hence

{1 \over { x }}<{{\int\limits_{1}^{x+h}{1 \over u}du -\int\limits_{1}^{x}{1 \over u}du} \over {h}} < {1 \over {x +h}},

from which (4) is obtained again.