add mathematical functions module

update docs and clean up code
2019-10-27 15:02:44 +01:00
parent b474188d1a
commit b566c643db
21 changed files with 592 additions and 71 deletions
--- a/docs/build/html/_modules/data.html
+++ b/docs/build/html/_modules/data.html
@@ -45,7 +45,6 @@

 <span class="sd">.. moduleauthor:: Daniel Weschke &lt;daniel.weschke@directbox.de&gt;</span>
 <span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">__future__</span> <span class="k">import</span> <span class="n">print_function</span>
 <span class="kn">import</span> <span class="nn">math</span>
 <span class="kn">import</span> <span class="nn">pickle</span>

@@ -79,7 +78,6 @@
    <span class="n">y</span> <span class="o">=</span> <span class="p">[]</span>
    <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">file</span><span class="p">:</span>
      <span class="n">fields</span> <span class="o">=</span> <span class="n">re</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;\s+&#39;</span><span class="p">,</span> <span class="n">row</span><span class="o">.</span><span class="n">strip</span><span class="p">())</span>
-      <span class="c1">#print(filds)</span>
      <span class="n">x</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">fields</span><span class="p">[</span><span class="n">x_column</span><span class="p">]))</span>
      <span class="n">y</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">fields</span><span class="p">[</span><span class="n">y_column</span><span class="p">]))</span>
    <span class="n">file</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
@@ -192,7 +190,6 @@
  <span class="n">exponent</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">start_exp</span><span class="p">,</span> <span class="n">step_exp</span><span class="p">)</span>  <span class="c1"># no stop because it is an open bound</span>

  <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">ceil</span><span class="p">((</span><span class="n">stop</span> <span class="o">-</span> <span class="n">start</span><span class="p">)</span><span class="o">/</span><span class="nb">float</span><span class="p">(</span><span class="n">step</span><span class="p">)))</span>
-  <span class="c1">#n = int((stop - start)/float(step))  # in py3 math.ceil returns int</span>
  <span class="n">lst</span> <span class="o">=</span> <span class="p">[]</span>
  <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
    <span class="n">lst</span> <span class="o">=</span> <span class="p">[</span><span class="nb">round</span><span class="p">(</span><span class="n">start</span> <span class="o">+</span> <span class="n">step</span><span class="o">*</span><span class="n">i</span><span class="p">,</span> <span class="n">exponent</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span>
@@ -227,7 +224,7 @@
 <span class="sd">  :returns: full id</span>
 <span class="sd">  :rtype: str or int</span>
 <span class="sd">  &quot;&quot;&quot;</span>
-  <span class="c1"># we know it is a unique ending</span>
+  <span class="c1"># take first element, because we know it is a unique ending</span>
  <span class="k">return</span> <span class="p">[</span><span class="n">idi</span> <span class="k">for</span> <span class="n">idi</span> <span class="ow">in</span> <span class="n">ids</span> <span class="k">if</span> <span class="n">idi</span><span class="o">.</span><span class="n">endswith</span><span class="p">(</span><span class="n">uide</span><span class="p">)][</span><span class="mi">0</span><span class="p">]</span></div>
 </pre></div>

--- a/docs/build/html/_modules/date.html
+++ b/docs/build/html/_modules/date.html
@@ -45,7 +45,6 @@
 <span class="sd">   </span>
 <span class="sd">.. moduleauthor:: Daniel Weschke &lt;daniel.weschke@directbox.de&gt;</span>
 <span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">__future__</span> <span class="k">import</span> <span class="n">division</span><span class="p">,</span> <span class="n">print_function</span><span class="p">,</span> <span class="n">unicode_literals</span>

 <div class="viewcode-block" id="gaußsche_osterformel"><a class="viewcode-back" href="../date.html#date.gaußsche_osterformel">[docs]</a><span class="k">def</span> <span class="nf">gaußsche_osterformel</span><span class="p">(</span><span class="n">year</span><span class="p">):</span>
  <span class="sd">&quot;&quot;&quot;Gaußsche Osterformel.</span>
--- a/docs/build/html/_modules/function.html
+++ b/docs/build/html/_modules/function.html
@@ -0,0 +1,199 @@
+
+<!DOCTYPE html>
+
+<html xmlns="http://www.w3.org/1999/xhtml">
+  <head>
+    <meta charset="utf-8" />
+    <title>function &#8212; pylib 2019.5.19 documentation</title>
+    <link rel="stylesheet" href="../_static/alabaster.css" type="text/css" />
+    <link rel="stylesheet" href="../_static/pygments.css" type="text/css" />
+    <link rel="stylesheet" href="../_static/custom.css" type="text/css" />
+    <script type="text/javascript" id="documentation_options" data-url_root="../" src="../_static/documentation_options.js"></script>
+    <script type="text/javascript" src="../_static/jquery.js"></script>
+    <script type="text/javascript" src="../_static/underscore.js"></script>
+    <script type="text/javascript" src="../_static/doctools.js"></script>
+    <script type="text/javascript" src="../_static/language_data.js"></script>
+    <script async="async" type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-AMS-MML_HTMLorMML"></script>
+    <link rel="index" title="Index" href="../genindex.html" />
+    <link rel="search" title="Search" href="../search.html" />
+   
+  <link rel="stylesheet" href="../_static/custom.css" type="text/css" />
+  
+  
+  <meta name="viewport" content="width=device-width, initial-scale=0.9, maximum-scale=0.9" />
+
+  </head><body>
+  
+
+    <div class="document">
+      <div class="documentwrapper">
+        <div class="bodywrapper">
+          
+
+          <div class="body" role="main">
+            
+  <h1>Source code for function</h1><div class="highlight"><pre>
+<span></span><span class="ch">#!/usr/bin/env python</span>
+<span class="c1"># -*- coding: utf-8 -*-</span>
+<span class="sd">&quot;&quot;&quot;Mathematical functions.</span>
+
+<span class="sd">:Date: 2019-10-27</span>
+
+<span class="sd">.. module:: function</span>
+<span class="sd">  :platform: *nix, Windows</span>
+<span class="sd">  :synopsis: Mathematical functions.</span>
+
+<span class="sd">.. moduleauthor:: Daniel Weschke &lt;daniel.weschke@directbox.de&gt;</span>
+<span class="sd">&quot;&quot;&quot;</span>
+<span class="kn">import</span> <span class="nn">math</span>
+
+<div class="viewcode-block" id="sine_wave"><a class="viewcode-back" href="../function.html#function.sine_wave">[docs]</a><span class="k">def</span> <span class="nf">sine_wave</span><span class="p">(</span><span class="n">A</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">f</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">phi</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">D</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+  <span class="sa">r</span><span class="sd">&quot;&quot;&quot;A sine wave or sinusoid is a mathematical curve that describes a</span>
+<span class="sd">  smooth periodic oscillation.</span>
+
+<span class="sd">  :param A: amplitude</span>
+<span class="sd">  :type A: float or int</span>
+<span class="sd">  :param k: (angular) wave number</span>
+<span class="sd">  :type k: float or int</span>
+<span class="sd">  :param f: ordinary frequency</span>
+<span class="sd">  :type f: float or int</span>
+<span class="sd">  :param phi: phase</span>
+<span class="sd">  :type phi: float or int</span>
+<span class="sd">  :param D: non-zero center amplitude</span>
+<span class="sd">  :type D: float or int</span>
+<span class="sd">  :param degree: boolean to switch between radians and degree. If</span>
+<span class="sd">    False phi is interpreted in radians and if True then phi is</span>
+<span class="sd">    interpreted in degrees.</span>
+
+<span class="sd">  In general, the function is:</span>
+
+<span class="sd">  .. math::</span>
+<span class="sd">    y(x,t) = A\sin(kx + 2\pi f t + \varphi) + D \\</span>
+<span class="sd">    y(x,t) = A\sin(kx + \omega t + \varphi) + D</span>
+
+<span class="sd">  where:</span>
+
+<span class="sd">    * A, amplitude, the peak deviation of the function from zero.</span>
+<span class="sd">    * f, ordinary frequency, the number of oscillations (cycles) that</span>
+<span class="sd">      occur each second of time.</span>
+<span class="sd">    * ω = 2πf, angular frequency, the rate of change of the function</span>
+<span class="sd">      argument in units of radians per second. If ω &lt; 0 the wave is</span>
+<span class="sd">      moving to the right, if ω &gt; 0 the wave is moving to the left.</span>
+<span class="sd">    * φ, phase, specifies (in radians) where in its cycle the</span>
+<span class="sd">      oscillation is at t = 0.</span>
+<span class="sd">    * x, spatial variable that represents the position on the</span>
+<span class="sd">      dimension on which the wave propagates.</span>
+<span class="sd">    * k, characteristic parameter called wave number (or angular wave</span>
+<span class="sd">      number), which represents the proportionality between the</span>
+<span class="sd">      angular frequency ω and the linear speed (speed of propagation)</span>
+<span class="sd">      ν.</span>
+<span class="sd">    * D, non-zero center amplitude.</span>
+
+<span class="sd">  The wavenumber is related to the angular frequency by:</span>
+
+<span class="sd">  .. math::</span>
+<span class="sd">    k={\omega \over v}={2\pi f \over v}={2\pi \over \lambda }</span>
+
+<span class="sd">  where λ (lambda) is the wavelength, f is the frequency, and v is the</span>
+<span class="sd">  linear speed.</span>
+
+<span class="sd">  .. seealso::</span>
+<span class="sd">    :meth:`function_cosine_wave_degree`</span>
+<span class="sd">  &quot;&quot;&quot;</span>
+  <span class="k">if</span> <span class="n">degree</span><span class="p">:</span>
+    <span class="n">phi</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">radians</span><span class="p">(</span><span class="n">phi</span><span class="p">)</span>
+  <span class="k">return</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="mi">0</span><span class="p">:</span> <span class="n">A</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">k</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">f</span><span class="o">*</span><span class="n">t</span> <span class="o">+</span> <span class="n">phi</span><span class="p">)</span> <span class="o">+</span> <span class="n">D</span></div>
+
+<div class="viewcode-block" id="cosine_wave"><a class="viewcode-back" href="../function.html#function.cosine_wave">[docs]</a><span class="k">def</span> <span class="nf">cosine_wave</span><span class="p">(</span><span class="n">A</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">f</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">phi</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">D</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+  <span class="sa">r</span><span class="sd">&quot;&quot;&quot;A cosine wave is said to be sinusoidal, because,</span>
+<span class="sd">  :math:`\cos(x) = \sin(x + \pi/2)`, which is also a sine wave with a</span>
+<span class="sd">  phase-shift of π/2 radians. Because of this head start, it is often</span>
+<span class="sd">  said that the cosine function leads the sine function or the sine</span>
+<span class="sd">  lags the cosine.</span>
+
+<span class="sd">  :param A: amplitude</span>
+<span class="sd">  :type A: float or int</span>
+<span class="sd">  :param k: (angular) wave number</span>
+<span class="sd">  :type k: float or int</span>
+<span class="sd">  :param f: ordinary frequency</span>
+<span class="sd">  :type f: float or int</span>
+<span class="sd">  :param phi: phase</span>
+<span class="sd">  :type phi: float or int</span>
+<span class="sd">  :param D: non-zero center amplitude</span>
+<span class="sd">  :type D: float or int</span>
+<span class="sd">  :param degree: boolean to switch between radians and degree. If</span>
+<span class="sd">    False phi is interpreted in radians and if True then phi is</span>
+<span class="sd">    interpreted in degrees.</span>
+
+<span class="sd">  .. seealso::</span>
+<span class="sd">    :meth:`function_sine_wave_degree`</span>
+<span class="sd">  &quot;&quot;&quot;</span>
+  <span class="k">if</span> <span class="n">degree</span><span class="p">:</span>
+    <span class="n">phi</span> <span class="o">=</span> <span class="n">phi</span> <span class="o">+</span> <span class="mi">90</span>
+  <span class="k">else</span><span class="p">:</span>
+    <span class="n">phi</span> <span class="o">=</span> <span class="n">phi</span> <span class="o">+</span> <span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span>
+  <span class="k">return</span> <span class="n">sine_wave</span><span class="p">(</span><span class="n">A</span><span class="o">=</span><span class="n">A</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="n">k</span><span class="p">,</span> <span class="n">f</span><span class="o">=</span><span class="n">f</span><span class="p">,</span> <span class="n">phi</span><span class="o">=</span><span class="n">phi</span><span class="p">,</span> <span class="n">D</span><span class="o">=</span><span class="n">D</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span></div>
+</pre></div>
+
+          </div>
+          
+        </div>
+      </div>
+      <div class="sphinxsidebar" role="navigation" aria-label="main navigation">
+        <div class="sphinxsidebarwrapper">
+<h1 class="logo"><a href="../index.html">pylib</a></h1>
+
+
+
+
+
+
+
+
+<h3>Navigation</h3>
+
+<div class="relations">
+<h3>Related Topics</h3>
+<ul>
+  <li><a href="../index.html">Documentation overview</a><ul>
+  <li><a href="index.html">Module code</a><ul>
+  </ul></li>
+  </ul></li>
+</ul>
+</div>
+<div id="searchbox" style="display: none" role="search">
+  <h3 id="searchlabel">Quick search</h3>
+    <div class="searchformwrapper">
+    <form class="search" action="../search.html" method="get">
+      <input type="text" name="q" aria-labelledby="searchlabel" />
+      <input type="submit" value="Go" />
+    </form>
+    </div>
+</div>
+<script type="text/javascript">$('#searchbox').show(0);</script>
+
+
+
+
+
+
+
+
+        </div>
+      </div>
+      <div class="clearer"></div>
+    </div>
+    <div class="footer">
+      &copy;2019, Daniel Weschke.
+      
+      |
+      Powered by <a href="http://sphinx-doc.org/">Sphinx 2.2.0</a>
+      &amp; <a href="https://github.com/bitprophet/alabaster">Alabaster 0.7.12</a>
+      
+    </div>
+
+    
+
+    
+  </body>
+</html>
--- a/docs/build/html/_modules/index.html
+++ b/docs/build/html/_modules/index.html
@@ -35,6 +35,7 @@
  <h1>All modules for which code is available</h1>
 <ul><li><a href="data.html">data</a></li>
 <li><a href="date.html">date</a></li>
+<li><a href="function.html">function</a></li>
 <li><a href="geometry.html">geometry</a></li>
 <li><a href="geometry_plot.html">geometry_plot</a></li>
 <li><a href="numerical/fit.html">numerical.fit</a></li>
--- a/docs/build/html/_modules/numerical/fit.html
+++ b/docs/build/html/_modules/numerical/fit.html
@@ -37,6 +37,8 @@
 <span class="c1"># -*- coding: utf-8 -*-</span>
 <span class="sd">&quot;&quot;&quot;Function and approximation.</span>

+<span class="sd">:Date: 2019-10-15</span>
+
 <span class="sd">.. module:: fit</span>
 <span class="sd">  :platform: *nix, Windows</span>
 <span class="sd">  :synopsis: Function and approximation.</span>
@@ -57,7 +59,7 @@
 <span class="sd">  :type x: int or float or list or numpy.ndarray</span>
 <span class="sd">  :param p: gauss parameters [a, b, c, d]:</span>

-<span class="sd">    * a -- amplitude (:math:`\int y \\,\\mathrm{d}x=1 \Leftrightarrow a=1/(c\\sqrt{2\\pi})` )</span>
+<span class="sd">    * a -- amplitude (:math:`\\int y \\,\\mathrm{d}x=1 \\Leftrightarrow a=1/(c\\sqrt{2\\pi})` )</span>
 <span class="sd">    * b -- expected value :math:`\\mu` (position of maximum, default = 0)</span>
 <span class="sd">    * c -- standard deviation :math:`\\sigma` (variance :math:`\\sigma^2=c^2`)</span>
 <span class="sd">    * d -- vertical offset (default = 0)</span>
@@ -127,7 +129,7 @@
  <span class="k">return</span> <span class="n">y_fit</span><span class="p">,</span> <span class="n">popt</span><span class="p">,</span> <span class="n">FWHM</span></div>

 <span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s2">&quot;__main__&quot;</span><span class="p">:</span>
-  <span class="kc">True</span>
+  <span class="k">pass</span>
 </pre></div>

          </div>
--- a/docs/build/html/_modules/numerical/integration.html
+++ b/docs/build/html/_modules/numerical/integration.html
@@ -51,7 +51,7 @@
 <span class="kn">from</span> <span class="nn">numpy</span> <span class="k">import</span> <span class="n">linspace</span><span class="p">,</span> <span class="n">trapz</span><span class="p">,</span> <span class="n">zeros</span>

 <div class="viewcode-block" id="trapez"><a class="viewcode-back" href="../../numerical.html#numerical.integration.trapez">[docs]</a><span class="k">def</span> <span class="nf">trapez</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
-  <span class="n">save_values</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
+           <span class="n">save_values</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
  <span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
 <span class="sd">  Integration of :math:`f(x)` using the trapezoidal rule</span>
 <span class="sd">  (Simpson&#39;s rule, Kepler&#39;s rule).</span>
@@ -122,14 +122,14 @@
 <span class="sd">    f(x) &amp;= x^2 \\</span>
 <span class="sd">    a &amp;= 0 \\</span>
 <span class="sd">    b &amp;= 1</span>
-<span class="sd">  </span>
+
 <span class="sd">  analytical solution</span>

 <span class="sd">  .. math::</span>
 <span class="sd">    I = \int\limits_{0}^{1} x^2 \,\mathrm{d}x</span>
 <span class="sd">      = \left. \frac{1}{3} x^3 \right\vert_0^1</span>
 <span class="sd">      = \frac{1}{3}</span>
-<span class="sd">  </span>
+
 <span class="sd">  numerical solution</span>

 <span class="sd">  &gt;&gt;&gt; f = lambda(x): x**2</span>
@@ -142,30 +142,30 @@
 <span class="sd">  &quot;&quot;&quot;</span>
  <span class="n">N</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
  <span class="c1"># f is function or list</span>
-  <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s1">&#39;__call__&#39;</span><span class="p">):</span>
-        <span class="c1"># h width of each subinterval</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">N</span>
+  <span class="k">if</span> <span class="n">callable</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
+    <span class="c1"># h width of each subinterval</span>
+    <span class="n">h</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">N</span>

-        <span class="c1"># x variable of integration</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">linspace</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">save_values</span><span class="p">:</span>
-            <span class="c1"># ff contribution from the points</span>
-            <span class="n">ff</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
-            <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
-                <span class="n">ff</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>
-            <span class="n">T</span> <span class="o">=</span> <span class="p">(</span><span class="n">ff</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="mf">2.</span><span class="o">+</span><span class="nb">sum</span><span class="p">(</span><span class="n">ff</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="n">N</span><span class="p">])</span><span class="o">+</span><span class="n">ff</span><span class="p">[</span><span class="n">N</span><span class="p">]</span><span class="o">/</span><span class="mf">2.</span><span class="p">)</span><span class="o">*</span><span class="n">h</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">TL</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="n">TR</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">N</span><span class="p">])</span>
-            <span class="n">TI</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="p">):</span>
-                <span class="n">TI</span> <span class="o">=</span> <span class="n">TI</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>
-            <span class="n">T</span> <span class="o">=</span> <span class="p">(</span><span class="n">TL</span><span class="o">/</span><span class="mf">2.</span><span class="o">+</span><span class="n">TI</span><span class="o">+</span><span class="n">TR</span><span class="o">/</span><span class="mf">2.</span><span class="p">)</span><span class="o">*</span><span class="n">h</span>
+    <span class="c1"># x variable of integration</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="n">linspace</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
+    <span class="k">if</span> <span class="n">save_values</span><span class="p">:</span>
+      <span class="c1"># ff contribution from the points</span>
+      <span class="n">ff</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
+      <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
+        <span class="n">ff</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>
+      <span class="n">T</span> <span class="o">=</span> <span class="p">(</span><span class="n">ff</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="mf">2.</span><span class="o">+</span><span class="nb">sum</span><span class="p">(</span><span class="n">ff</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="n">N</span><span class="p">])</span><span class="o">+</span><span class="n">ff</span><span class="p">[</span><span class="n">N</span><span class="p">]</span><span class="o">/</span><span class="mf">2.</span><span class="p">)</span><span class="o">*</span><span class="n">h</span>
+    <span class="k">else</span><span class="p">:</span>
+      <span class="n">TL</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+      <span class="n">TR</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">N</span><span class="p">])</span>
+      <span class="n">TI</span> <span class="o">=</span> <span class="mi">0</span>
+      <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="p">):</span>
+        <span class="n">TI</span> <span class="o">=</span> <span class="n">TI</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>
+      <span class="n">T</span> <span class="o">=</span> <span class="p">(</span><span class="n">TL</span><span class="o">/</span><span class="mf">2.</span><span class="o">+</span><span class="n">TI</span><span class="o">+</span><span class="n">TR</span><span class="o">/</span><span class="mf">2.</span><span class="p">)</span><span class="o">*</span><span class="n">h</span>
  <span class="k">else</span><span class="p">:</span>
-        <span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span>
-        <span class="n">T</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
-            <span class="n">T</span> <span class="o">=</span> <span class="n">T</span> <span class="o">+</span> <span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="p">])</span><span class="o">/</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="n">f</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>
+    <span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span>
+    <span class="n">T</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
+      <span class="n">T</span> <span class="o">=</span> <span class="n">T</span> <span class="o">+</span> <span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="p">])</span><span class="o">/</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="n">f</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>

  <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
    <span class="nb">print</span><span class="p">(</span><span class="n">T</span><span class="p">)</span>
@@ -178,12 +178,12 @@
  <span class="n">trapez</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">1e6</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
  <span class="c1">#print(trapz(func, linspace(0,1,10)))</span>

-  <span class="n">trapez</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">9</span><span class="p">],</span> <span class="n">x</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+  <span class="n">trapez</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">9</span><span class="p">],</span> <span class="n">x</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
  <span class="c1">#print(trapz([0,1,4,9]))</span>

-  <span class="n">trapez</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="n">x</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+  <span class="n">trapez</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="n">x</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>

-  <span class="n">trapez</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">x</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+  <span class="n">trapez</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">x</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
 </pre></div>

          </div>
--- a/docs/build/html/_modules/time_of_day.html
+++ b/docs/build/html/_modules/time_of_day.html
@@ -45,10 +45,8 @@
 <span class="sd">   </span>
 <span class="sd">.. moduleauthor:: Daniel Weschke &lt;daniel.weschke@directbox.de&gt;</span>
 <span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">__future__</span> <span class="k">import</span> <span class="n">division</span><span class="p">,</span> <span class="n">print_function</span><span class="p">,</span> <span class="n">unicode_literals</span>
 <span class="kn">from</span> <span class="nn">time</span> <span class="k">import</span> <span class="n">struct_time</span><span class="p">,</span> <span class="n">mktime</span>

-
 <div class="viewcode-block" id="in_seconds"><a class="viewcode-back" href="../time_of_day.html#time_of_day.in_seconds">[docs]</a><span class="k">def</span> <span class="nf">in_seconds</span><span class="p">(</span><span class="n">time</span><span class="p">):</span>
  <span class="sd">&quot;&quot;&quot;If time is `time.struct_time` convert to float seconds.</span>

--- a/docs/build/html/_sources/function.rst.txt
+++ b/docs/build/html/_sources/function.rst.txt
@@ -0,0 +1,7 @@
+function module
+===============
+
+.. automodule:: function
+   :members:
+   :undoc-members:
+   :show-inheritance:
--- a/docs/build/html/_sources/modules.rst.txt
+++ b/docs/build/html/_sources/modules.rst.txt
@@ -6,6 +6,7 @@ pylib

   data
   date
+   function
   geometry
   geometry_plot
   numerical
--- a/docs/build/html/function.html
+++ b/docs/build/html/function.html
@@ -0,0 +1,190 @@
+
+<!DOCTYPE html>
+
+<html xmlns="http://www.w3.org/1999/xhtml">
+  <head>
+    <meta charset="utf-8" />
+    <title>function module &#8212; pylib 2019.5.19 documentation</title>
+    <link rel="stylesheet" href="_static/alabaster.css" type="text/css" />
+    <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
+    <link rel="stylesheet" href="_static/custom.css" type="text/css" />
+    <script type="text/javascript" id="documentation_options" data-url_root="./" src="_static/documentation_options.js"></script>
+    <script type="text/javascript" src="_static/jquery.js"></script>
+    <script type="text/javascript" src="_static/underscore.js"></script>
+    <script type="text/javascript" src="_static/doctools.js"></script>
+    <script type="text/javascript" src="_static/language_data.js"></script>
+    <script async="async" type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-AMS-MML_HTMLorMML"></script>
+    <link rel="index" title="Index" href="genindex.html" />
+    <link rel="search" title="Search" href="search.html" />
+   
+  <link rel="stylesheet" href="_static/custom.css" type="text/css" />
+  
+  
+  <meta name="viewport" content="width=device-width, initial-scale=0.9, maximum-scale=0.9" />
+
+  </head><body>
+  
+
+    <div class="document">
+      <div class="documentwrapper">
+        <div class="bodywrapper">
+          
+
+          <div class="body" role="main">
+            
+  <div class="section" id="module-function">
+<span id="function-module"></span><h1>function module<a class="headerlink" href="#module-function" title="Permalink to this headline">¶</a></h1>
+<p>Mathematical functions.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Date</dt>
+<dd class="field-odd"><p>2019-10-27</p>
+</dd>
+</dl>
+<span class="target" id="module-function"></span><dl class="function">
+<dt id="function.cosine_wave">
+<code class="sig-name descname">cosine_wave</code><span class="sig-paren">(</span><em class="sig-param">A=1</em>, <em class="sig-param">k=1</em>, <em class="sig-param">f=1</em>, <em class="sig-param">phi=0</em>, <em class="sig-param">D=0</em>, <em class="sig-param">degree=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/function.html#cosine_wave"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#function.cosine_wave" title="Permalink to this definition">¶</a></dt>
+<dd><p>A cosine wave is said to be sinusoidal, because,
+<span class="math notranslate nohighlight">\(\cos(x) = \sin(x + \pi/2)\)</span>, which is also a sine wave with a
+phase-shift of π/2 radians. Because of this head start, it is often
+said that the cosine function leads the sine function or the sine
+lags the cosine.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters</dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>A</strong> (<em>float</em><em> or </em><em>int</em>) – amplitude</p></li>
+<li><p><strong>k</strong> (<em>float</em><em> or </em><em>int</em>) – (angular) wave number</p></li>
+<li><p><strong>f</strong> (<em>float</em><em> or </em><em>int</em>) – ordinary frequency</p></li>
+<li><p><strong>phi</strong> (<em>float</em><em> or </em><em>int</em>) – phase</p></li>
+<li><p><strong>D</strong> (<em>float</em><em> or </em><em>int</em>) – non-zero center amplitude</p></li>
+<li><p><strong>degree</strong> – boolean to switch between radians and degree. If
+False phi is interpreted in radians and if True then phi is
+interpreted in degrees.</p></li>
+</ul>
+</dd>
+</dl>
+<div class="admonition seealso">
+<p class="admonition-title">See also</p>
+<p><code class="xref py py-meth docutils literal notranslate"><span class="pre">function_sine_wave_degree()</span></code></p>
+</div>
+</dd></dl>
+
+<dl class="function">
+<dt id="function.sine_wave">
+<code class="sig-name descname">sine_wave</code><span class="sig-paren">(</span><em class="sig-param">A=1</em>, <em class="sig-param">k=1</em>, <em class="sig-param">f=1</em>, <em class="sig-param">phi=0</em>, <em class="sig-param">D=0</em>, <em class="sig-param">degree=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/function.html#sine_wave"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#function.sine_wave" title="Permalink to this definition">¶</a></dt>
+<dd><p>A sine wave or sinusoid is a mathematical curve that describes a
+smooth periodic oscillation.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters</dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>A</strong> (<em>float</em><em> or </em><em>int</em>) – amplitude</p></li>
+<li><p><strong>k</strong> (<em>float</em><em> or </em><em>int</em>) – (angular) wave number</p></li>
+<li><p><strong>f</strong> (<em>float</em><em> or </em><em>int</em>) – ordinary frequency</p></li>
+<li><p><strong>phi</strong> (<em>float</em><em> or </em><em>int</em>) – phase</p></li>
+<li><p><strong>D</strong> (<em>float</em><em> or </em><em>int</em>) – non-zero center amplitude</p></li>
+<li><p><strong>degree</strong> – boolean to switch between radians and degree. If
+False phi is interpreted in radians and if True then phi is
+interpreted in degrees.</p></li>
+</ul>
+</dd>
+</dl>
+<p>In general, the function is:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}y(x,t) = A\sin(kx + 2\pi f t + \varphi) + D \\
+y(x,t) = A\sin(kx + \omega t + \varphi) + D\end{split}\]</div>
+<p>where:</p>
+<blockquote>
+<div><ul class="simple">
+<li><p>A, amplitude, the peak deviation of the function from zero.</p></li>
+<li><p>f, ordinary frequency, the number of oscillations (cycles) that
+occur each second of time.</p></li>
+<li><p>ω = 2πf, angular frequency, the rate of change of the function
+argument in units of radians per second. If ω &lt; 0 the wave is
+moving to the right, if ω &gt; 0 the wave is moving to the left.</p></li>
+<li><p>φ, phase, specifies (in radians) where in its cycle the
+oscillation is at t = 0.</p></li>
+<li><p>x, spatial variable that represents the position on the
+dimension on which the wave propagates.</p></li>
+<li><p>k, characteristic parameter called wave number (or angular wave
+number), which represents the proportionality between the
+angular frequency ω and the linear speed (speed of propagation)
+ν.</p></li>
+<li><p>D, non-zero center amplitude.</p></li>
+</ul>
+</div></blockquote>
+<p>The wavenumber is related to the angular frequency by:</p>
+<div class="math notranslate nohighlight">
+\[k={\omega \over v}={2\pi f \over v}={2\pi \over \lambda }\]</div>
+<p>where λ (lambda) is the wavelength, f is the frequency, and v is the
+linear speed.</p>
+<div class="admonition seealso">
+<p class="admonition-title">See also</p>
+<p><code class="xref py py-meth docutils literal notranslate"><span class="pre">function_cosine_wave_degree()</span></code></p>
+</div>
+</dd></dl>
+
+</div>
+
+
+          </div>
+          
+        </div>
+      </div>
+      <div class="sphinxsidebar" role="navigation" aria-label="main navigation">
+        <div class="sphinxsidebarwrapper">
+<h1 class="logo"><a href="index.html">pylib</a></h1>
+
+
+
+
+
+
+
+
+<h3>Navigation</h3>
+
+<div class="relations">
+<h3>Related Topics</h3>
+<ul>
+  <li><a href="index.html">Documentation overview</a><ul>
+  </ul></li>
+</ul>
+</div>
+<div id="searchbox" style="display: none" role="search">
+  <h3 id="searchlabel">Quick search</h3>
+    <div class="searchformwrapper">
+    <form class="search" action="search.html" method="get">
+      <input type="text" name="q" aria-labelledby="searchlabel" />
+      <input type="submit" value="Go" />
+    </form>
+    </div>
+</div>
+<script type="text/javascript">$('#searchbox').show(0);</script>
+
+
+
+
+
+
+
+
+        </div>
+      </div>
+      <div class="clearer"></div>
+    </div>
+    <div class="footer">
+      &copy;2019, Daniel Weschke.
+      
+      |
+      Powered by <a href="http://sphinx-doc.org/">Sphinx 2.2.0</a>
+      &amp; <a href="https://github.com/bitprophet/alabaster">Alabaster 0.7.12</a>
+      
+      |
+      <a href="_sources/function.rst.txt"
+          rel="nofollow">Page source</a>
+    </div>
+
+    
+
+    
+  </body>
+</html>
--- a/docs/build/html/genindex.html
+++ b/docs/build/html/genindex.html
@@ -72,6 +72,8 @@
 <h2 id="C">C</h2>
 <table style="width: 100%" class="indextable genindextable"><tr>
  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="function.html#function.cosine_wave">cosine_wave() (in module function)</a>
+</li>
      <li><a href="geometry.html#geometry.cubic">cubic() (in module geometry)</a>
 </li>
  </ul></td>
@@ -132,11 +134,13 @@
  <td style="width: 33%; vertical-align: top;"><ul>
      <li><a href="numerical.html#module-fit">fit (module)</a>
 </li>
-  </ul></td>
-  <td style="width: 33%; vertical-align: top;"><ul>
      <li><a href="data.html#data.fold_list">fold_list() (in module data)</a>
 </li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
      <li><a href="numerical.html#numerical.ode.fpi">fpi() (in module numerical.ode)</a>
+</li>
+      <li><a href="function.html#module-function">function (module)</a>, <a href="function.html#module-function">[1]</a>
 </li>
  </ul></td>
 </tr></table>
@@ -287,10 +291,12 @@
      <li><a href="time_of_day.html#time_of_day.seconds">seconds() (in module time_of_day)</a>
 </li>
      <li><a href="time_of_day.html#time_of_day.seconds_norm">seconds_norm() (in module time_of_day)</a>
+</li>
+      <li><a href="data.html#data.seq">seq() (in module data)</a>
 </li>
  </ul></td>
  <td style="width: 33%; vertical-align: top;"><ul>
-      <li><a href="data.html#data.seq">seq() (in module data)</a>
+      <li><a href="function.html#function.sine_wave">sine_wave() (in module function)</a>
 </li>
      <li><a href="geometry.html#geometry.square">square() (in module geometry)</a>
 </li>
--- a/docs/build/html/modules.html
+++ b/docs/build/html/modules.html
@@ -38,6 +38,7 @@
 <ul>
 <li class="toctree-l1"><a class="reference internal" href="data.html">data module</a></li>
 <li class="toctree-l1"><a class="reference internal" href="date.html">date module</a></li>
+<li class="toctree-l1"><a class="reference internal" href="function.html">function module</a></li>
 <li class="toctree-l1"><a class="reference internal" href="geometry.html">geometry module</a></li>
 <li class="toctree-l1"><a class="reference internal" href="geometry_plot.html">geometry_plot module</a></li>
 <li class="toctree-l1"><a class="reference internal" href="numerical.html">numerical package</a><ul>
--- a/docs/build/html/numerical.html
+++ b/docs/build/html/numerical.html
@@ -40,6 +40,11 @@
 <div class="section" id="module-numerical.fit">
 <span id="numerical-fit-module"></span><h2>numerical.fit module<a class="headerlink" href="#module-numerical.fit" title="Permalink to this headline">¶</a></h2>
 <p>Function and approximation.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Date</dt>
+<dd class="field-odd"><p>2019-10-15</p>
+</dd>
+</dl>
 <span class="target" id="module-fit"></span><dl class="function">
 <dt id="numerical.fit.gauss">
 <code class="sig-name descname">gauss</code><span class="sig-paren">(</span><em class="sig-param">x</em>, <em class="sig-param">*p</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/numerical/fit.html#gauss"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numerical.fit.gauss" title="Permalink to this definition">¶</a></dt>
--- a/docs/build/html/objects.inv
+++ b/docs/build/html/objects.inv
--- a/docs/build/html/py-modindex.html
+++ b/docs/build/html/py-modindex.html
@@ -74,6 +74,11 @@
       <td>
       <a href="numerical.html#module-fit"><code class="xref">fit</code></a> <em>(*nix, Windows)</em></td><td>
       <em>Function and approximation.</em></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="function.html#module-function"><code class="xref">function</code></a> <em>(*nix, Windows)</em></td><td>
+       <em>Mathematical functions.</em></td></tr>
     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
     <tr class="cap" id="cap-g"><td></td><td>
       <strong>g</strong></td><td></td></tr>
--- a/docs/build/html/searchindex.js
+++ b/docs/build/html/searchindex.js
--- a/docs/source/function.rst
+++ b/docs/source/function.rst
@@ -0,0 +1,7 @@
+function module
+===============
+
+.. automodule:: function
+   :members:
+   :undoc-members:
+   :show-inheritance:
--- a/docs/source/modules.rst
+++ b/docs/source/modules.rst
@@ -6,6 +6,7 @@ pylib

   data
   date
+   function
   geometry
   geometry_plot
   numerical
--- a/pylib/function.py
+++ b/pylib/function.py
@@ -0,0 +1,100 @@
+#!/usr/bin/env python
+# -*- coding: utf-8 -*-
+"""Mathematical functions.
+
+:Date: 2019-10-27
+
+.. module:: function
+  :platform: *nix, Windows
+  :synopsis: Mathematical functions.
+
+.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
+"""
+import math
+
+def sine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False):
+  r"""A sine wave or sinusoid is a mathematical curve that describes a
+  smooth periodic oscillation.
+
+  :param A: amplitude
+  :type A: float or int
+  :param k: (angular) wave number
+  :type k: float or int
+  :param f: ordinary frequency
+  :type f: float or int
+  :param phi: phase
+  :type phi: float or int
+  :param D: non-zero center amplitude
+  :type D: float or int
+  :param degree: boolean to switch between radians and degree. If
+    False phi is interpreted in radians and if True then phi is
+    interpreted in degrees.
+
+  In general, the function is:
+
+  .. math::
+    y(x,t) = A\sin(kx + 2\pi f t + \varphi) + D \\
+    y(x,t) = A\sin(kx + \omega t + \varphi) + D
+
+  where:
+
+    * A, amplitude, the peak deviation of the function from zero.
+    * f, ordinary frequency, the number of oscillations (cycles) that
+      occur each second of time.
+    * ω = 2πf, angular frequency, the rate of change of the function
+      argument in units of radians per second. If ω < 0 the wave is
+      moving to the right, if ω > 0 the wave is moving to the left.
+    * φ, phase, specifies (in radians) where in its cycle the
+      oscillation is at t = 0.
+    * x, spatial variable that represents the position on the
+      dimension on which the wave propagates.
+    * k, characteristic parameter called wave number (or angular wave
+      number), which represents the proportionality between the
+      angular frequency ω and the linear speed (speed of propagation)
+      ν.
+    * D, non-zero center amplitude.
+
+  The wavenumber is related to the angular frequency by:
+
+  .. math::
+    k={\omega \over v}={2\pi f \over v}={2\pi \over \lambda }
+
+  where λ (lambda) is the wavelength, f is the frequency, and v is the
+  linear speed.
+
+  .. seealso::
+    :meth:`function_cosine_wave_degree`
+  """
+  if degree:
+    phi = math.radians(phi)
+  return lambda x, t=0: A*math.sin(k*x + 2*math.pi*f*t + phi) + D
+
+def cosine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False):
+  r"""A cosine wave is said to be sinusoidal, because,
+  :math:`\cos(x) = \sin(x + \pi/2)`, which is also a sine wave with a
+  phase-shift of π/2 radians. Because of this head start, it is often
+  said that the cosine function leads the sine function or the sine
+  lags the cosine.
+
+  :param A: amplitude
+  :type A: float or int
+  :param k: (angular) wave number
+  :type k: float or int
+  :param f: ordinary frequency
+  :type f: float or int
+  :param phi: phase
+  :type phi: float or int
+  :param D: non-zero center amplitude
+  :type D: float or int
+  :param degree: boolean to switch between radians and degree. If
+    False phi is interpreted in radians and if True then phi is
+    interpreted in degrees.
+
+  .. seealso::
+    :meth:`function_sine_wave_degree`
+  """
+  if degree:
+    phi = phi + 90
+  else:
+    phi = phi + math.pi/2
+  return sine_wave(A=A, k=k, f=f, phi=phi, D=D, degree=degree)
--- a/pylib/numerical/fit.py
+++ b/pylib/numerical/fit.py
@@ -2,6 +2,8 @@
 # -*- coding: utf-8 -*-
 """Function and approximation.

+:Date: 2019-10-15
+
 .. module:: fit
  :platform: *nix, Windows
  :synopsis: Function and approximation.
@@ -22,7 +24,7 @@ def gauss(x, *p):
  :type x: int or float or list or numpy.ndarray
  :param p: gauss parameters [a, b, c, d]:

-    * a -- amplitude (:math:`\int y \\,\\mathrm{d}x=1 \Leftrightarrow a=1/(c\\sqrt{2\\pi})` )
+    * a -- amplitude (:math:`\\int y \\,\\mathrm{d}x=1 \\Leftrightarrow a=1/(c\\sqrt{2\\pi})` )
    * b -- expected value :math:`\\mu` (position of maximum, default = 0)
    * c -- standard deviation :math:`\\sigma` (variance :math:`\\sigma^2=c^2`)
    * d -- vertical offset (default = 0)
@@ -92,4 +94,4 @@ def gauss_fit(x, y, e=None, x_fit=None, verbose=False):
  return y_fit, popt, FWHM

 if __name__ == "__main__":
-  True
+  pass
--- a/pylib/numerical/integration.py
+++ b/pylib/numerical/integration.py
@@ -16,7 +16,7 @@ from __future__ import division
 from numpy import linspace, trapz, zeros

 def trapez(f, a=0, b=1, N=10, x=None, verbose=False,
-  save_values=False):
+           save_values=False):
  r"""
  Integration of :math:`f(x)` using the trapezoidal rule
  (Simpson's rule, Kepler's rule).
@@ -87,14 +87,14 @@ def trapez(f, a=0, b=1, N=10, x=None, verbose=False,
    f(x) &= x^2 \\
    a &= 0 \\
    b &= 1
-  
+
  analytical solution

  .. math::
    I = \int\limits_{0}^{1} x^2 \,\mathrm{d}x
      = \left. \frac{1}{3} x^3 \right\vert_0^1
      = \frac{1}{3}
-  
+
  numerical solution

  >>> f = lambda(x): x**2
@@ -107,30 +107,30 @@ def trapez(f, a=0, b=1, N=10, x=None, verbose=False,
  """
  N = int(N)
  # f is function or list
-  if hasattr(f, '__call__'):
-        # h width of each subinterval
-        h = (b-a)/N
+  if callable(f):
+    # h width of each subinterval
+    h = (b-a)/N

-        # x variable of integration
-        x = linspace(a, b, N+1)
-        if save_values:
-            # ff contribution from the points
-            ff = zeros((N+1))
-            for n in linspace(0, N, N+1):
-                ff[n] = f(x[n])
-            T = (ff[0]/2.+sum(ff[1:N])+ff[N]/2.)*h
-        else:
-            TL = f(x[0])
-            TR = f(x[N])
-            TI = 0
-            for n in range(1, N):
-                TI = TI + f(x[n])
-            T = (TL/2.+TI+TR/2.)*h
+    # x variable of integration
+    x = linspace(a, b, N+1)
+    if save_values:
+      # ff contribution from the points
+      ff = zeros((N+1))
+      for n in linspace(0, N, N+1):
+        ff[n] = f(x[n])
+      T = (ff[0]/2.+sum(ff[1:N])+ff[N]/2.)*h
+    else:
+      TL = f(x[0])
+      TR = f(x[N])
+      TI = 0
+      for n in range(1, N):
+        TI = TI + f(x[n])
+      T = (TL/2.+TI+TR/2.)*h
  else:
-        N = len(f)-1
-        T = 0
-        for n in range(N):
-            T = T + (x[n+1]-x[n])/2*(f[n+1]+f[n])
+    N = len(f)-1
+    T = 0
+    for n in range(N):
+      T = T + (x[n+1]-x[n])/2*(f[n+1]+f[n])

  if verbose:
    print(T)
@@ -143,9 +143,9 @@ if __name__ == '__main__':
  trapez(func, 0, 1, 1e6, verbose=True)
  #print(trapz(func, linspace(0,1,10)))

-  trapez([0,1,4,9], x=[0,1,2,3], verbose=True)
+  trapez([0, 1, 4, 9], x=[0, 1, 2, 3], verbose=True)
  #print(trapz([0,1,4,9]))

-  trapez([2,2], x=[-1,1], verbose=True)
+  trapez([2, 2], x=[-1, 1], verbose=True)

-  trapez([-1,1,1,-1], x=[-1,-1,1,1], verbose=True)
+  trapez([-1, 1, 1, -1], x=[-1, -1, 1, 1], verbose=True)