.. _chap-more-taylor-series:

********************
 More taylor series
********************

.. _sec-starting-to-study-exponentials:

Exponentials
============


The number :math:`e`: base for natural exponentials and logarithms
------------------------------------------------------------------

In earlier lessons we discussed the exponential function at length.
Here we discuss some of the reasons for which we consider Euler's
number :math:`e` to be the "natural" base for exponentials and
logarithms, analogously to how radians are the natural unit of measure
for circles.

Part of this is a visual exploration of :math:`e^x` and an graphical
look at its slope, to then conclude that:

.. math::

   \frac{d e^x}{dx} = e^x

which is not true for other bases, like :math:`2^x` and :math:`10^x`

We can also mention that:

.. math::

   e = \lim_{n \rightarrow \infty} \left(1 + \frac{1}{n}\right)^n

and calculate it for values of n like 1, 10, 100, 1000, 10000, ...  We
can also calculate :math:`e` with:

.. math::

   e = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} +
   \frac{1}{4!} + \frac{1}{5!} + \dots = \sum_{k=0}^{\infty}
   \frac{1}{k!} \approx 2.71828182846

but the justification for that will only come when we show how to
calculate the Taylor series for :math:`e^x`.


.. caution::

   It is also worth mentioning that difference communities in math,
   science, and engineering use :math:`\log(x)` to mean the *natural*
   logarithm (base :math:`e`).  Other communities will use
   :math:`\log(x)` to mean the *base 10* logarithm, and they say
   :math:`\ln(x)` for the natural logarithm.  I use the former
   approach, and the various programming language libraries (C,
   Python, ...) do the same.



But we take a break from these notes as we get to use a proper text
book to explore exponentials and logarithms in more detail.


The Taylor series for :math:`e^x` -- visually
---------------------------------------------

First we look at it visually.  We will want to plot terms in the
following way in gnuplot and geogebra and desmos:

.. code-block:: console

   $ gnuplot
   ## then the following lines have the prompt gnuplot> and we type:
   reset
   set grid
   set xrange [-1:3]
   set terminal qt linewidth 3
   plot exp(x) lw 2
   replot 1
   replot 1 + x
   replot 1 + x + x**2 / 2!
   replot 1 + x + x**2 / 2! + x**3 / 3!
   replot 1 + x + x**2 / 2! + x**3 / 3! + x**4 / 4!
   replot 1 + x + x**2 / 2! + x**3 / 3! + x**4 / 4! + x**5 / 5!
   replot 1 + x + x**2 / 2! + x**3 / 3! + x**4 / 4! + x**5 / 5! + x**6 / 6!

In geogebra or desmos:

::

   e^x
   1
   1 + x
   1 + x + x**2 / 2!
   1 + x + x**2 / 2! + x**3 / 3!
   1 + x + x**2 / 2! + x**3 / 3! + x**4 / 4!
   1 + x + x**2 / 2! + x**3 / 3! + x**4 / 4! + x**5 / 5!
   1 + x + x**2 / 2! + x**3 / 3! + x**4 / 4! + x**5 / 5! + x**6 / 6!
   

The Taylor series for :math:`e^x` -- calculating the coefficients
-----------------------------------------------------------------

Remembering that :math:`\frac{de^x}{dx} = e^x` we can calculate:

.. math::

   \frac{d^k e^x}{dx^k} \bigg \rvert_{x = 0} = e^0 = 1

which gives us Taylor coefficients :math:`a_k = \frac{1}{k!}` and the
series looks like:

.. math::

   e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}


Miscellaneous Taylor expansions
===============================

Logarithms:

.. math::

   log(1 - x) = -\sum_{k=1}^{\infty} \frac{x^k}{k} \\
   log(1 + x) = \sum_{k=1}^{\infty} (-1)^{k+1} \frac{x^k}{k}

the first when :math:`|x| < 1`, the second when :math:`-1 < x \leq 1`

Note that when you plot these logarithmic functions you will need to
double check that your plotting program uses :math:`log()` for
*natural* logarithms.  Some plotting programs use :math:`\ln()` for
natural logarithms.

Geometric series:

.. math::
   :nowrap:

   \begin{eqnarray}
   \frac{1}{1-x}      = & \sum_{k=0}^{\infty} & x^k \\
   \frac{1}{(1-x)^2}  = & \sum_{k=1}^{\infty} & k x^{k-1} \\
   \frac{1}{(1-x)^3}  = & \sum_{k=2}^{\infty} & \frac{(k-1)k}{2} x^{k-2}
   \end{eqnarray}

when :math:`|x| < 1`


Some square root expansions
===========================

Square root functions can get complicated.  For example, the
relativistic formula for the rest *plus* kinetic energy of an object
with mass :math:`m_0` is

.. math::

   E_{\rm total} = \frac{m_0 c^2}{\sqrt{1 - \frac{v^2}{c^2}}}

This has the famous Lorenz gamma factor:

.. math::

   \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}

We sometimes use a shorthand :math:`\beta = v/c`, where :math:`\beta`
is the velocity expressed as a *fraction of the speed of light*, and
get:

.. math::

   \gamma = \frac{1}{\sqrt{1 - \beta^2}}


The first few terms in the taylor series expansion in :math:`\beta`
are (see the Cupcake Physics link in the resources chapter for
details):

..
   .. math::

      \gamma \;\; = \; & 1 & + \frac{1}{2}\beta^2         & + & \frac{3}{8}\beta^4         & + & \frac{5}{16}\beta^6         & + & \frac{35}{128}\beta^8 \dots\\
                  = \; & 1 & + \frac{1}{2}\frac{v^2}{c^2} & + & \frac{3}{8}\frac{v^4}{c^4} & + & \frac{5}{16}\frac{v^6}{c^6} & + & \frac{35}{128}\frac{v^8}{c^8}\dots

.. math::
   :nowrap:

   \begin{eqnarray}
   \gamma \;\; = \; & 1  + \frac{1}{2}\beta^2          +  \frac{3}{8}\beta^4          +  \frac{5}{16}\beta^6          +  \frac{35}{128}\beta^8 \dots\\
               = \; & 1  + \frac{1}{2}\frac{v^2}{c^2}  +  \frac{3}{8}\frac{v^4}{c^4}  +  \frac{5}{16}\frac{v^6}{c^6}  + & \frac{35}{128}\frac{v^8}{c^8}\dots
    \end{eqnarray}

Putting this back into the formula for energy we get:

.. math::

   E_\textrm{total} = \; \frac{m_0 c^2}{\sqrt{1 - \frac{v^2}{c^2}}}
                 = \; m_0 c^2 + \frac{1}{2} m_0 v^2 + \dots

For low values of :math:`v^2/c^2` (i.e. :math:`v` much slower than the
speed of light) we have:

.. math::

   E_{\rm total} = m_0 c^2 + \frac{1}{2} m_0 v^2 + \dots

We can read off the terms and realize that the total energy is equal
to the famous rest mass :math:`E_{\rm rest} = m_0 c^2` plus the
kinetic energy :math:`\frac{1}{2} m_0 v^2 + \dots`:

.. math::

   E_{\rm total} = E_{\rm rest} + E_{\rm kinetic} = m_0 c^2 +
   \frac{1}{2} m_0 v^2
   + \frac{3}{8} m_0 \frac{v^4}{c^2} \dots

Let us explore the Lorenz gamma factor for values of :math:`v` in the
whole range from 0 to :math:`c`:

::

   $ gnuplot
   ## then the following lines have the prompt gnuplot> and we type:
   reset
   set grid
   set ylabel '\gamma'
   set xlabel '\beta (v/c)'
   set xrange [0:1]
   set terminal qt linewidth 3
   plot 1 / sqrt(1 - x**2)

Or in a web-based graphing calculator:

::

   1 / (1 - x^2)^(1/2)

.. _fig-lorenz_factor:

.. figure:: lorenz_factor.*
   :width: 60%

   The lorenz factor as a function of :math:`\beta = v/c`.  Note
   how it is close to 1 for most of the run, but grows out of control
   when :math:`v` approaches the speed of light :math:`c`.


What insight does this give us on the energy of an object as it
approaches the speed of light?  Note that the formulae for length and
time are:

.. math::

   L = \frac{1}{\gamma} L_0 \\
   \Delta t' = \gamma \Delta t

so the behavior of :math:`\gamma` as a function of :math:`\beta` (and
thus :math:`v`) also affects length and time.

Now let us look at the polynomial approximates in :math:`\beta`:

::

   $ gnuplot
   ## then the following lines have the prompt gnuplot> and we type:
   reset
   set grid
   set ylabel '\gamma'
   set xlabel '\beta (v^2/c^2)'
   set xrange [0:0.0001]
   set terminal qt linewidth 3
   plot 1 / (1 - x**2)
   replot 1
   replot 1 + (1.0/2) * x**2
   replot 1 + (1.0/2) * x**2 + (3.0/8) * x**4 
   replot 1 + (1.0/2) * x**2 + (3.0/8) * x**4 + (5.0/16) * x**6