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<h1 class="chapter"> 23. Numerical </h1>

<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC87">23.1 Introduction to fast Fourier transform</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">                     
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC88">23.2 Functions and Variables for fast Fourier transform</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC89">23.3 Introduction to Fourier series</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC90">23.4 Functions and Variables for Fourier series</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
</table>

<p><a name="Item_003a-Introduction-to-fast-Fourier-transform"></a>
</p><hr size="6">
<a name="Introduction-to-fast-Fourier-transform"></a>
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</tr></table>
<h2 class="section"> 23.1 Introduction to fast Fourier transform </h2>

<p>The <code>fft</code> package comprises functions for the numerical (not symbolic) computation
of the fast Fourier transform.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Fourier-transform">Fourier transform</a>
 &middot;
<a href="maxima_95.html#Category_003a-Numerical-methods">Numerical methods</a>
 &middot;
<a href="maxima_95.html#Category_003a-Share-packages">Share packages</a>
 &middot;
<a href="maxima_95.html#Category_003a-Package-fft">Package fft</a>
</p>
</div>




<p><a name="Item_003a-Functions-and-Variables-for-fast-Fourier-transform"></a>
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</tr></table>
<h2 class="section"> 23.2 Functions and Variables for fast Fourier transform </h2>

<p><a name="Item_003a-polartorect"></a>
</p><dl>
<dt><u>Function:</u> <b>polartorect</b><i> (<var>r</var>, <var>t</var>)</i>
<a name="IDX788"></a>
</dt>
<dd><p>Translates complex values of the form <code>r %e^(%i t)</code> to the form <code>a + b %i</code>,
where <var>r</var> is the magnitude and <var>t</var> is the phase.
<var>r</var> and <var>t</var> are 1-dimensional arrays of the same size.
The array size need not be a power of 2.
</p>
<p>The original values of the input arrays are
replaced by the real and imaginary parts, <code>a</code> and <code>b</code>, on return.
The outputs are calculated as
</p>
<pre class="example">a = r cos(t)
b = r sin(t)
</pre>
<p><code>polartorect</code> is the inverse function of <code>recttopolar</code>.
</p>
<p><code>load(fft)</code> loads this function. See also <code>fft</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-fft">Package fft</a>
 &middot;
<a href="maxima_95.html#Category_003a-Complex-variables">Complex variables</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-recttopolar"></a>
</p><dl>
<dt><u>Function:</u> <b>recttopolar</b><i> (<var>a</var>, <var>b</var>)</i>
<a name="IDX789"></a>
</dt>
<dd><p>Translates complex values of the form <code>a + b %i</code> to the form <code>r %e^(%i t)</code>,
where <var>a</var> is the real part and <var>b</var> is the imaginary part.
<var>a</var> and <var>b</var> are 1-dimensional arrays of the same size.
The array size need not be a power of 2.
</p>
<p>The original values of the input arrays are
replaced by the magnitude and angle, <code>r</code> and <code>t</code>, on return.
The outputs are calculated as
</p>
<pre class="example">r = sqrt(a^2 + b^2)
t = atan2(b, a)
</pre>
<p>The computed angle is in the range <code>-%pi</code> to <code>%pi</code>. 
</p>
<p><code>recttopolar</code> is the inverse function of <code>polartorect</code>.
</p>
<p><code>load(fft)</code> loads this function. See also <code>fft</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-fft">Package fft</a>
 &middot;
<a href="maxima_95.html#Category_003a-Complex-variables">Complex variables</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-inverse_005ffft"></a>
</p><dl>
<dt><u>Function:</u> <b>inverse_fft</b><i> (<var>y</var>)</i>
<a name="IDX790"></a>
</dt>
<dd><p>Computes the inverse complex fast Fourier transform.
<var>y</var> is a list or array (named or unnamed) which contains the data to transform.
The number of elements must be a power of 2.
The elements must be literal numbers (integers, rationals, floats, or bigfloats)
or symbolic constants,
or expressions <code>a + b*%i</code> where <code>a</code> and <code>b</code> are literal numbers
or symbolic constants.
</p>
<p><code>inverse_fft</code> returns a new object of the same type as <var>y</var>,
which is not modified.
Results are always computed as floats
or expressions <code>a + b*%i</code> where <code>a</code> and <code>b</code> are floats.
</p>
<p>The inverse discrete Fourier transform is defined as follows.
Let <code>x</code> be the output of the inverse transform.
Then for <code>j</code> from 0 through <code>n - 1</code>,
</p>
<pre class="example">x[j] = sum(y[k] exp(+2 %i %pi j k / n), k, 0, n - 1)
</pre>
<p><code>load(fft)</code> loads this function.
</p>
<p>See also <code>fft</code> (forward transform), <code>recttopolar</code>, and <code>polartorect</code>.
</p>
<p>Examples:
</p>
<p>Real data.
</p>
<pre class="example">(%i1) load (fft) $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $
(%i4) L1 : inverse_fft (L);
(%o4) [0.0, 14.49 %i - .8284, 0.0, 2.485 %i + 4.828, 0.0, 
                       4.828 - 2.485 %i, 0.0, - 14.49 %i - .8284]
(%i5) L2 : fft (L1);
(%o5) [1.0, 2.0 - 2.168L-19 %i, 3.0 - 7.525L-20 %i, 
4.0 - 4.256L-19 %i, - 1.0, 2.168L-19 %i - 2.0, 
7.525L-20 %i - 3.0, 4.256L-19 %i - 4.0]
(%i6) lmax (abs (L2 - L));
(%o6)                       3.545L-16
</pre>
<p>Complex data.
</p>
<pre class="example">(%i1) load (fft) $
(%i2) fpprintprec : 4 $                 
(%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $
(%i4) L1 : inverse_fft (L);
(%o4) [4.0, 2.711L-19 %i + 4.0, 2.0 %i - 2.0, 
- 2.828 %i - 2.828, 0.0, 5.421L-20 %i + 4.0, - 2.0 %i - 2.0, 
2.828 %i + 2.828]
(%i5) L2 : fft (L1);
(%o5) [4.066E-20 %i + 1.0, 1.0 %i + 1.0, 1.0 - 1.0 %i, 
1.55L-19 %i - 1.0, - 4.066E-20 %i - 1.0, 1.0 - 1.0 %i, 
1.0 %i + 1.0, 1.0 - 7.368L-20 %i]
(%i6) lmax (abs (L2 - L));                    
(%o6)                       6.841L-17
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-fft">Package fft</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-fft"></a>
</p><dl>
<dt><u>Function:</u> <b>fft</b><i> (<var>x</var>)</i>
<a name="IDX791"></a>
</dt>
<dd><p>Computes the complex fast Fourier transform.
<var>x</var> is a list or array (named or unnamed) which contains the data to transform.
The number of elements must be a power of 2.
The elements must be literal numbers (integers, rationals, floats, or bigfloats)
or symbolic constants,
or expressions <code>a + b*%i</code> where <code>a</code> and <code>b</code> are literal numbers
or symbolic constants.
</p>
<p><code>fft</code> returns a new object of the same type as <var>x</var>,
which is not modified.
Results are always computed as floats
or expressions <code>a + b*%i</code> where <code>a</code> and <code>b</code> are floats.
</p>
<p>The discrete Fourier transform is defined as follows.
Let <code>y</code> be the output of the transform.
Then for <code>k</code> from 0 through <code>n - 1</code>,
</p>
<pre class="example">y[k] = (1/n) sum(x[j] exp(-2 %i %pi j k / n), j, 0, n - 1)
</pre>
<p>When the data <var>x</var> are real,
real coefficients <code>a</code> and <code>b</code> can be computed such that
</p>
<pre class="example">x[j] = sum (a[k] * cos (2*%pi*j*k / n) + b[k] * sin (2*%pi*j*k / n), k, 0, n/2)
</pre>
<p>with
</p>
<pre class="example">a[0] = realpart (y[0])
b[0] = 0
</pre>
<p>and, for k from 1 through n/2 - 1,
</p>
<pre class="example">a[k] = realpart (y[k] + y[n - k])
b[k] = imagpart (y[n - k] - y[k])
</pre>
<p>and
</p>
<pre class="example">a[n/2] = realpart (y[n/2])
b[n/2] = 0
</pre>
<p><code>load(fft)</code> loads this function.
</p>
<p>See also <code>inverse_fft</code> (inverse transform), <code>recttopolar</code>, and <code>polartorect</code>.
</p>
<p>Examples:
</p>
<p>Real data.
</p>
<pre class="example">(%i1) load (fft) $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $
(%i4) L1 : fft (L);
(%o4) [0.0, - 1.811 %i - .1036, 0.0, .6036 - .3107 %i, 0.0, 
                         .3107 %i + .6036, 0.0, 1.811 %i - .1036]
(%i5) L2 : inverse_fft (L1);
(%o5) [1.0, 2.168L-19 %i + 2.0, 7.525L-20 %i + 3.0, 
4.256L-19 %i + 4.0, - 1.0, - 2.168L-19 %i - 2.0, 
- 7.525L-20 %i - 3.0, - 4.256L-19 %i - 4.0]
(%i6) lmax (abs (L2 - L));
(%o6)                       3.545L-16
</pre>
<p>Complex data.
</p>
<pre class="example">(%i1) load (fft) $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $
(%i4) L1 : fft (L);
(%o4) [0.5, .3536 %i + .3536, - 0.25 %i - 0.25, 
0.5 - 6.776L-21 %i, 0.0, - .3536 %i - .3536, 0.25 %i - 0.25, 
0.5 - 3.388L-20 %i]
(%i5) L2 : inverse_fft (L1);
(%o5) [1.0 - 4.066E-20 %i, 1.0 %i + 1.0, 1.0 - 1.0 %i, 
- 1.008L-19 %i - 1.0, 4.066E-20 %i - 1.0, 1.0 - 1.0 %i, 
1.0 %i + 1.0, 1.947L-20 %i + 1.0]
(%i6) lmax (abs (L2 - L));
(%o6)                       6.83L-17
</pre>
<p>Computation of sine and cosine coefficients.
</p>
<pre class="example">(%i1) load (fft) $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 2, 3, 4, 5, 6, 7, 8] $
(%i4) n : length (L) $
(%i5) x : make_array (any, n) $
(%i6) fillarray (x, L) $
(%i7) y : fft (x) $
(%i8) a : make_array (any, n/2 + 1) $
(%i9) b : make_array (any, n/2 + 1) $
(%i10) a[0] : realpart (y[0]) $
(%i11) b[0] : 0 $
(%i12) for k : 1 thru n/2 - 1 do
   (a[k] : realpart (y[k] + y[n - k]),
    b[k] : imagpart (y[n - k] - y[k]));
(%o12)                        done
(%i13) a[n/2] : y[n/2] $
(%i14) b[n/2] : 0 $
(%i15) listarray (a);
(%o15)          [4.5, - 1.0, - 1.0, - 1.0, - 0.5]
(%i16) listarray (b);
(%o16)           [0, - 2.414, - 1.0, - .4142, 0]
(%i17) f(j) := sum (a[k] * cos (2*%pi*j*k / n) + b[k] * sin (2*%pi*j*k / n), k, 0, n/2) $
(%i18) makelist (float (f (j)), j, 0, n - 1);
(%o18)      [1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0]
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-fft">Package fft</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-fortindent"></a>
</p><dl>
<dt><u>Option variable:</u> <b>fortindent</b>
<a name="IDX792"></a>
</dt>
<dd><p>Default value: 0
</p>
<p><code>fortindent</code> controls the left margin indentation of
expressions printed out by the <code>fortran</code> command.  0 gives normal
printout (i.e., 6 spaces), and positive values will causes the
expressions to be printed farther to the right.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Translation-and-compilation">Translation and compilation</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-fortran"></a>
</p><dl>
<dt><u>Function:</u> <b>fortran</b><i> (<var>expr</var>)</i>
<a name="IDX793"></a>
</dt>
<dd><p>Prints <var>expr</var> as a Fortran statement.
The output line is indented with spaces.
If the line is too long, <code>fortran</code> prints continuation lines.
<code>fortran</code> prints the exponentiation operator <code>^</code> as <code>**</code>,
and prints a complex number <code>a + b %i</code> in the form <code>(a,b)</code>.
</p>
<p><var>expr</var> may be an equation. If so, <code>fortran</code> prints an assignment
statement, assigning the right-hand side of the equation to the left-hand side.
In particular, if the right-hand side of <var>expr</var> is the name of a matrix,
then <code>fortran</code> prints an assignment statement for each element of the matrix.
</p>
<p>If <var>expr</var> is not something recognized by <code>fortran</code>,
the expression is printed in <code>grind</code> format without complaint.
<code>fortran</code> does not know about lists, arrays, or functions.
</p>
<p><code>fortindent</code> controls the left margin of the printed lines.
0 is the normal margin (i.e., indented 6 spaces). Increasing <code>fortindent</code>
causes expressions to be printed further to the right.
</p>
<p>When <code>fortspaces</code> is <code>true</code>, <code>fortran</code> fills out
each printed line with spaces to 80 columns.
</p>
<p><code>fortran</code> evaluates its arguments;
quoting an argument defeats evaluation.
<code>fortran</code> always returns <code>done</code>.
</p>
<p>Examples:
</p>
<pre class="verbatim">(%i1) expr: (a + b)^12$
(%i2) fortran (expr);
      (b+a)**12                                                                 
(%o2)                         done
(%i3) fortran ('x=expr);
      x = (b+a)**12                                                             
(%o3)                         done
(%i4) fortran ('x=expand (expr));
      x = b**12+12*a*b**11+66*a**2*b**10+220*a**3*b**9+495*a**4*b**8+792
     1   *a**5*b**7+924*a**6*b**6+792*a**7*b**5+495*a**8*b**4+220*a**9*b
     2   **3+66*a**10*b**2+12*a**11*b+a**12
(%o4)                         done
(%i5) fortran ('x=7+5*%i);
      x = (7,5)                                                                 
(%o5)                         done
(%i6) fortran ('x=[1,2,3,4]);
      x = [1,2,3,4]                                                             
(%o6)                         done
(%i7) f(x) := x^2$
(%i8) fortran (f);
      f                                                                         
(%o8)                         done
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Translation-and-compilation">Translation and compilation</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-fortspaces"></a>
</p><dl>
<dt><u>Option variable:</u> <b>fortspaces</b>
<a name="IDX794"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>fortspaces</code> is <code>true</code>, <code>fortran</code> fills out
each printed line with spaces to 80 columns.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Translation-and-compilation">Translation and compilation</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-horner"></a>
</p><dl>
<dt><u>Function:</u> <b>horner</b><i> (<var>expr</var>, <var>x</var>)</i>
<a name="IDX795"></a>
</dt>
<dt><u>Function:</u> <b>horner</b><i> (<var>expr</var>)</i>
<a name="IDX796"></a>
</dt>
<dd><p>Returns a rearranged representation of <var>expr</var> as
in Horner's rule, using <var>x</var> as the main variable if it is specified.
<code>x</code> may be omitted in which case the main variable of the canonical rational expression
form of <var>expr</var> is used.
</p>
<p><code>horner</code> sometimes improves stability if <code>expr</code> is
to be numerically evaluated.  It is also useful if Maxima is used to
generate programs to be run in Fortran. See also <code>stringout</code>.
</p>
<pre class="example">(%i1) expr: 1e-155*x^2 - 5.5*x + 5.2e155;
                           2
(%o1)            1.0E-155 x  - 5.5 x + 5.2E+155
(%i2) expr2: horner (%, x), keepfloat: true;
(%o2)            (1.0E-155 x - 5.5) x + 5.2E+155
(%i3) ev (expr, x=1e155);
Maxima encountered a Lisp error:

 floating point overflow

Automatically continuing.
To reenable the Lisp debugger set *debugger-hook* to nil.
(%i4) ev (expr2, x=1e155);
(%o4)                       7.0E+154
</pre>
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<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Numerical-methods">Numerical methods</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-find_005froot"></a>
</p><dl>
<dt><u>Function:</u> <b>find_root</b><i> (<var>expr</var>, <var>x</var>, <var>a</var>, <var>b</var>)</i>
<a name="IDX797"></a>
</dt>
<dt><u>Function:</u> <b>find_root</b><i> (<var>f</var>, <var>a</var>, <var>b</var>)</i>
<a name="IDX798"></a>
</dt>
<dt><u>Option variable:</u> <b>find_root_error</b>
<a name="IDX799"></a>
</dt>
<dt><u>Option variable:</u> <b>find_root_abs</b>
<a name="IDX800"></a>
</dt>
<dt><u>Option variable:</u> <b>find_root_rel</b>
<a name="IDX801"></a>
</dt>
<dd><p>Finds a root of the expression <var>expr</var> or the function <var>f</var>
over the closed interval <em>[<var>a</var>, <var>b</var>]</em>.
The expression <var>expr</var> may be an equation,
in which case <code>find_root</code> seeks a root of <code>lhs(<var>expr</var>) - rhs(<var>expr</var>)</code>.
</p>
<p>Given that Maxima can evaluate <var>expr</var> or <var>f</var> over <em>[<var>a</var>, <var>b</var>]</em>
and that <var>expr</var> or <var>f</var> is continuous,
<code>find_root</code> is guaranteed to find the root,
or one of the roots if there is more than one.
</p>
<p><code>find_root</code> initially applies binary search.
If the function in question appears to be smooth enough,
<code>find_root</code> applies linear interpolation instead.
</p>
<p>The accuracy of <code>find_root</code> is governed by <code>find_root_abs</code> and <code>find_root_rel</code>.
<code>find_root</code> stops when the function in question
evaluates to something less than or equal to <code>find_root_abs</code>,
or if successive approximants <var>x_0</var>, <var>x_1</var> differ by no more than
<code>find_root_rel * max(abs(x_0), abs(x_1))</code>.
The default values of <code>find_root_abs</code> and <code>find_root_rel</code> are both zero.
</p>
<p><code>find_root</code> expects the function in question to have a different sign at the endpoints
of the search interval.
When the function evaluates to a number at both endpoints
and these numbers have the same sign,
the behavior of <code>find_root</code> is governed by <code>find_root_error</code>.
When <code>find_root_error</code> is <code>true</code>,
<code>find_root</code> prints an error message.
Otherwise <code>find_root</code> returns the value of <code>find_root_error</code>.
The default value of <code>find_root_error</code> is <code>true</code>.
</p>
<p>If <var>f</var> evaluates to something other than a number at any step in the search algorithm,
<code>find_root</code> returns a partially-evaluated <code>find_root</code> expression.
</p>
<p>The order of <var>a</var> and <var>b</var> is ignored;
the region in which a root is sought is <em>[min(<var>a</var>, <var>b</var>), max(<var>a</var>, <var>b</var>)]</em>.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) f(x) := sin(x) - x/2;
                                        x
(%o1)                  f(x) := sin(x) - -
                                        2
(%i2) find_root (sin(x) - x/2, x, 0.1, %pi);
(%o2)                   1.895494267033981
(%i3) find_root (sin(x) = x/2, x, 0.1, %pi);
(%o3)                   1.895494267033981
(%i4) find_root (f(x), x, 0.1, %pi);
(%o4)                   1.895494267033981
(%i5) find_root (f, 0.1, %pi);
(%o5)                   1.895494267033981
(%i6) find_root (exp(x) = y, x, 0, 100);
                            x
(%o6)           find_root(%e  = y, x, 0.0, 100.0)
(%i7) find_root (exp(x) = y, x, 0, 100), y = 10;
(%o7)                   2.302585092994046
(%i8) log (10.0);
(%o8)                   2.302585092994046
</pre>
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<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Algebraic-equations">Algebraic equations</a>
 &middot;
<a href="maxima_95.html#Category_003a-Numerical-methods">Numerical methods</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-newton"></a>
</p><dl>
<dt><u>Function:</u> <b>newton</b><i> (<var>expr</var>, <var>x</var>, <var>x_0</var>, <var>eps</var>)</i>
<a name="IDX802"></a>
</dt>
<dd><p>Returns an approximate solution of <code><var>expr</var> = 0</code> by Newton's method,
considering <var>expr</var> to be a function of one variable, <var>x</var>.
The search begins with <code><var>x</var> = <var>x_0</var></code>
and proceeds until <code>abs(<var>expr</var>) &lt; <var>eps</var></code>
(with <var>expr</var> evaluated at the current value of <var>x</var>).
</p>
<p><code>newton</code> allows undefined variables to appear in <var>expr</var>,
so long as the termination test <code>abs(<var>expr</var>) &lt; <var>eps</var></code> evaluates
to <code>true</code> or <code>false</code>.
Thus it is not necessary that <var>expr</var> evaluate to a number.
</p>
<p><code>load(newton1)</code> loads this function.
</p>
<p>See also <code>realroots</code>, <code>allroots</code>, <code>find_root</code>, and <code>mnewton</code>.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) load (newton1);
(%o1) /usr/share/maxima/5.10.0cvs/share/numeric/newton1.mac
(%i2) newton (cos (u), u, 1, 1/100);
(%o2)                   1.570675277161251
(%i3) ev (cos (u), u = %);
(%o3)                 1.2104963335033528E-4
(%i4) assume (a &gt; 0);
(%o4)                        [a &gt; 0]
(%i5) newton (x^2 - a^2, x, a/2, a^2/100);
(%o5)                  1.00030487804878 a
(%i6) ev (x^2 - a^2, x = %);
                                           2
(%o6)                6.098490481853958E-4 a
</pre>
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<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Algebraic-equations">Algebraic equations</a>
 &middot;
<a href="maxima_95.html#Category_003a-Numerical-methods">Numerical methods</a>
</p>
</div>


</dd></dl>


<p><a name="Item_003a-Introduction-to-Fourier-series"></a>
</p><hr size="6">
<a name="Introduction-to-Fourier-series"></a>
<a name="SEC89"></a>
<table cellpadding="1" cellspacing="1" border="0">
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<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<h2 class="section"> 23.3 Introduction to Fourier series </h2>

<p>The <code>fourie</code> package comprises functions for the symbolic computation
of Fourier series.
There are functions in the <code>fourie</code> package to calculate Fourier integral
coefficients and some functions for manipulation of expressions.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Fourier-transform">Fourier transform</a>
 &middot;
<a href="maxima_95.html#Category_003a-Share-packages">Share packages</a>
 &middot;
<a href="maxima_95.html#Category_003a-Package-fourie">Package fourie</a>
</p>
</div>



<p><a name="Item_003a-Functions-and-Variables-for-Fourier-series"></a>
</p><hr size="6">
<a name="Functions-and-Variables-for-Fourier-series"></a>
<a name="SEC90"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#SEC89" title="Previous section in reading order"> &lt; </a>]</td>
<td valign="middle" align="left">[<a href="maxima_24.html#SEC91" title="Next section in reading order"> &gt; </a>]</td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left">[<a href="#SEC86" title="Beginning of this chapter or previous chapter"> &lt;&lt; </a>]</td>
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<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
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<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<h2 class="section"> 23.4 Functions and Variables for Fourier series </h2>

<p><a name="Item_003a-equalp"></a>
</p><dl>
<dt><u>Function:</u> <b>equalp</b><i> (<var>x</var>, <var>y</var>)</i>
<a name="IDX803"></a>
</dt>
<dd><p>Returns <code>true</code> if <code>equal (<var>x</var>, <var>y</var>)</code> otherwise <code>false</code> (doesn't give an
error message like <code>equal (x, y)</code> would do in this case).
</p>
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</p>
</div>

</dd></dl>

<p><a name="Item_003a-remfun"></a>
</p><dl>
<dt><u>Function:</u> <b>remfun</b><i> (<var>f</var>, <var>expr</var>)</i>
<a name="IDX804"></a>
</dt>
<dt><u>Function:</u> <b>remfun</b><i> (<var>f</var>, <var>expr</var>, <var>x</var>)</i>
<a name="IDX805"></a>
</dt>
<dd><p><code>remfun (<var>f</var>, <var>expr</var>)</code>
replaces all occurrences of <code><var>f</var> (<var>arg</var>)</code> by <var>arg</var> in <var>expr</var>.
</p>
<p><code>remfun (<var>f</var>, <var>expr</var>, <var>x</var>)</code>
replaces all occurrences of <code><var>f</var> (<var>arg</var>)</code> by <var>arg</var> in <var>expr</var>
only if <var>arg</var> contains the variable <var>x</var>.
</p>
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</p>
</div>

</dd></dl>

<p><a name="Item_003a-funp"></a>
</p><dl>
<dt><u>Function:</u> <b>funp</b><i> (<var>f</var>, <var>expr</var>)</i>
<a name="IDX806"></a>
</dt>
<dt><u>Function:</u> <b>funp</b><i> (<var>f</var>, <var>expr</var>, <var>x</var>)</i>
<a name="IDX807"></a>
</dt>
<dd><p><code>funp (<var>f</var>, <var>expr</var>)</code>
returns <code>true</code> if <var>expr</var> contains the function <var>f</var>.
</p>
<p><code>funp (<var>f</var>, <var>expr</var>, <var>x</var>)</code>
returns <code>true</code> if <var>expr</var> contains the function <var>f</var> and the variable
<var>x</var> is somewhere in the argument of one of the instances of <var>f</var>.
</p>
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</p>
</div>

</dd></dl>

<p><a name="Item_003a-absint"></a>
</p><dl>
<dt><u>Function:</u> <b>absint</b><i> (<var>f</var>, <var>x</var>, <var>halfplane</var>)</i>
<a name="IDX808"></a>
</dt>
<dt><u>Function:</u> <b>absint</b><i> (<var>f</var>, <var>x</var>)</i>
<a name="IDX809"></a>
</dt>
<dt><u>Function:</u> <b>absint</b><i> (<var>f</var>, <var>x</var>, <var>a</var>, <var>b</var>)</i>
<a name="IDX810"></a>
</dt>
<dd><p><code>absint (<var>f</var>, <var>x</var>, <var>halfplane</var>)</code>
returns the indefinite integral of <var>f</var> with respect to
<var>x</var> in the given halfplane (<code>pos</code>, <code>neg</code>, or <code>both</code>).
<var>f</var> may contain expressions of the form
<code>abs (x)</code>, <code>abs (sin (x))</code>, <code>abs (a) * exp (-abs (b) * abs (x))</code>.
</p>
<p><code>absint (<var>f</var>, <var>x</var>)</code> is equivalent to <code>absint (<var>f</var>, <var>x</var>, pos)</code>.
</p>
<p><code>absint (<var>f</var>, <var>x</var>, <var>a</var>, <var>b</var>)</code>
returns the definite integral of <var>f</var> with respect to <var>x</var> from <var>a</var> to <var>b</var>.
<var>f</var> may include absolute values.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-fourie">Package fourie</a>
 &middot;
<a href="maxima_95.html#Category_003a-Integral-calculus">Integral calculus</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-fourier"></a>
</p><dl>
<dt><u>Function:</u> <b>fourier</b><i> (<var>f</var>, <var>x</var>, <var>p</var>)</i>
<a name="IDX811"></a>
</dt>
<dd><p>Returns a list of the Fourier coefficients of <code><var>f</var>(<var>x</var>)</code> defined
on the interval <code>[-p, p]</code>.
</p>
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</p>
</div>

</dd></dl>

<p><a name="Item_003a-foursimp"></a>
</p><dl>
<dt><u>Function:</u> <b>foursimp</b><i> (<var>l</var>)</i>
<a name="IDX812"></a>
</dt>
<dd><p>Simplifies <code>sin (n %pi)</code> to 0 if <code>sinnpiflag</code> is <code>true</code> and
<code>cos (n %pi)</code> to <code>(-1)^n</code> if <code>cosnpiflag</code> is <code>true</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-fourie">Package fourie</a>
 &middot;
<a href="maxima_95.html#Category_003a-Trigonometric-functions">Trigonometric functions</a>
 &middot;
<a href="maxima_95.html#Category_003a-Simplification-functions">Simplification functions</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-sinnpiflag"></a>
</p><dl>
<dt><u>Option variable:</u> <b>sinnpiflag</b>
<a name="IDX813"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>See <code>foursimp</code>.
</p>
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</p>
</div>


</dd></dl>

<p><a name="Item_003a-cosnpiflag"></a>
</p><dl>
<dt><u>Option variable:</u> <b>cosnpiflag</b>
<a name="IDX814"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>See <code>foursimp</code>.
</p>
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</p>
</div>


</dd></dl>

<p><a name="Item_003a-fourexpand"></a>
</p><dl>
<dt><u>Function:</u> <b>fourexpand</b><i> (<var>l</var>, <var>x</var>, <var>p</var>, <var>limit</var>)</i>
<a name="IDX815"></a>
</dt>
<dd><p>Constructs and returns the Fourier series from the list of
Fourier coefficients <var>l</var> up through <var>limit</var> terms (<var>limit</var>
may be <code>inf</code>). <var>x</var> and <var>p</var> have same meaning as in
<code>fourier</code>.
</p>
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</p>
</div>

</dd></dl>

<p><a name="Item_003a-fourcos"></a>
</p><dl>
<dt><u>Function:</u> <b>fourcos</b><i> (<var>f</var>, <var>x</var>, <var>p</var>)</i>
<a name="IDX816"></a>
</dt>
<dd><p>Returns the Fourier cosine coefficients for <code><var>f</var>(<var>x</var>)</code> defined on <code>[0, <var>p</var>]</code>.
</p>
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</p>
</div>

</dd></dl>

<p><a name="Item_003a-foursin"></a>
</p><dl>
<dt><u>Function:</u> <b>foursin</b><i> (<var>f</var>, <var>x</var>, <var>p</var>)</i>
<a name="IDX817"></a>
</dt>
<dd><p>Returns the Fourier sine coefficients for <code><var>f</var>(<var>x</var>)</code> defined on <code>[0, <var>p</var>]</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-fourie">Package fourie</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-totalfourier"></a>
</p><dl>
<dt><u>Function:</u> <b>totalfourier</b><i> (<var>f</var>, <var>x</var>, <var>p</var>)</i>
<a name="IDX818"></a>
</dt>
<dd><p>Returns <code>fourexpand (foursimp (fourier (<var>f</var>, <var>x</var>, <var>p</var>)), <var>x</var>, <var>p</var>, 'inf)</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-fourie">Package fourie</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-fourint"></a>
</p><dl>
<dt><u>Function:</u> <b>fourint</b><i> (<var>f</var>, <var>x</var>)</i>
<a name="IDX819"></a>
</dt>
<dd><p>Constructs and returns a list of the Fourier integral coefficients of <code><var>f</var>(<var>x</var>)</code>
defined on <code>[minf, inf]</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-fourie">Package fourie</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-fourintcos"></a>
</p><dl>
<dt><u>Function:</u> <b>fourintcos</b><i> (<var>f</var>, <var>x</var>)</i>
<a name="IDX820"></a>
</dt>
<dd><p>Returns the Fourier cosine integral coefficients for <code><var>f</var>(<var>x</var>)</code> on <code>[0, inf]</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-fourie">Package fourie</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-fourintsin"></a>
</p><dl>
<dt><u>Function:</u> <b>fourintsin</b><i> (<var>f</var>, <var>x</var>)</i>
<a name="IDX821"></a>
</dt>
<dd><p>Returns the Fourier sine integral coefficients for <code><var>f</var>(<var>x</var>)</code> on <code>[0, inf]</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-fourie">Package fourie</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-Arrays"></a>
</p><hr size="6">
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