Examples by topic
=================

Differentiation
---------------
The operator ``df`` is used to represent partial differentiation with respect 
to one or more variables. 

.. pull-quote:: ``syntax: df exprn [var <num>]+.`` 

Differentiation of the function :math:`x^2 y^3 z^4` with respect to 
:math:`x,y,z`, two, three and four times respectively, i.e
:math:`\frac{\partial^9 x^2 y^3 z^4}{\partial x^2 \partial y^3 \partial z^4}`:

.. code:: python

  > simplify $ df (x^2*y^3*z^4) x 2 y 3 z 4 ;
  288
  
The derivative of :math:`\log \sin (x)^2`:

.. code:: python

  > simplify $ df (log(sin x)^2) x;
  2*cos x*log (sin x)/sin x

  Note the parentheses.


Suppose :math:`z(\cos(x),y)`: Let's calculate :math:`\frac{\partial \sin(z)}
{\partial \cos(x)}` and :math:`\frac{\partial z^2}{\partial x}` : 

.. code:: python 

  > declare depend [z,cos x,y];
  []
  > simplify (df (sin z) (cos x));
  cos z*df z (cos x)
  > simplify (df (z^2) x);
  2*df z x*z

  Note how to declare dependencies.
  
The results are :math:`\cos(z) \frac{\partial z}
{\partial \cos(x)}` and  :math:`2 z \frac{\partial z} {\partial x})`
respectively, as expected.

Integration
-----------
``INT`` is an operator in ``REDUCE`` for indefinite integration using a 
combination of the Risch-Norman algorithm and pattern matching. 

.. pull-quote:: ``syntax: intg exprn var.``

Note that in ``Pure`` the operator is called ``intg`` in order not to clash 
with the integer type ``int``.

Example 1:

.. math:: \int \frac{1}{a x + b} dx 

.. code:: python

  > simplify $ intg (1/(a*x+b)) x;
  log (a*x+b)/a

Example 2:

.. math:: I(a,b,n) = \int x^2 (a x + b)^n dx 

.. code:: python

  > I a b n = simplify $ intg (x^2*(a*x+b)^n) x;
  > I a b n;
  ((a*x+b)^n*a^3*n^2*x^3+3*(a*x+b)^n*a^3*n*x^3+2*(a*x+b)^n*a^3*x^3+
   (a*x+b)^n*a^2*b*n^2*x^2+(a*x+b)^n*a^2*b*n*x^2-2*(a*x+b)^n*a*b^2*
    n*x+2*(a*x+b)^n*b^3)/(a^3*n^3+6*a^3*n^2+11*a^3*n+6*a^3)
  > I a b 0 ;
  x^3/3
  > I 0 b n;
  b^n*x^3/3
  > I a 0 k;
  x^k*a^k*x^3/(k+3)
  
  
Example 3: 
  
.. math:: \int \frac{\sqrt{x+\sqrt{x^2+1}}}{x}

.. code:: python

  > simplify $ intg (sqrt(x+sqrt(x^2+1))/x) x ;
  intg (sqrt (sqrt (x^2+1)+x)/x) x
  
Apparently no solution was found. There is a package ``ALGINT`` in REDUCE, that
is specialized to deal with algebraic functions. The [UserGuide]_ says

.. pull-quote:: ... *will analytically integrate a wide range of expressions 
   involving square roots where the answer exists in that class of functions. 
   It is an implementation of the work described in J.H. Davenport*
   [LNCS102]_ 

.. code:: python

   > reduce::load "algint" ;
   0
   > simplify $ intg (sqrt(x+sqrt(x^2+1))/x) x ;
   atan ((sqrt (sqrt (x^2+1)+x)*sqrt (x^2+1)-sqrt (sqrt (x^2+1)+x)*x-sqrt 
   (sqrt (x^2+1)+x))/2)+2*sqrt (sqrt (x^2+1)+x)+log (sqrt (sqrt 
   (x^2+1)+x)-1)-log (sqrt (sqrt (x^2+1)+x)+1)

   Note how to load packages.
   
.. [UserGuide] REDUCE User Guide
   ref. 
.. [LNCS102] On the Integration of Algebraic Functions, LNCS 102, 
   Springer Verlag, 1981.