322 lines
16 KiB
HTML
322 lines
16 KiB
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<!DOCTYPE HTML PUBLIC "-//W3O//DTD W3 HTML 2.0//EN">
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<!Converted with LaTeX2HTML 0.6.5 (Tue Nov 15 1994) by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds >
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<HEAD>
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<TITLE>12.5.2. Trigonometric and Related Functions</TITLE>
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</HEAD>
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<BODY>
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<meta name="description" value=" Trigonometric and Related Functions">
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<meta name="keywords" value="clm">
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<meta name="resource-type" value="document">
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<meta name="distribution" value="global">
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<P>
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<b>Common Lisp the Language, 2nd Edition</b>
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<BR> <HR><A NAME=tex2html3129 HREF="node129.html"><IMG ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME=tex2html3127 HREF="node126.html"><IMG ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME=tex2html3121 HREF="node127.html"><IMG ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <A NAME=tex2html3131 HREF="node1.html"><IMG ALIGN=BOTTOM ALT="contents" SRC="icons/contents_motif.gif"></A> <A NAME=tex2html3132 HREF="index.html"><IMG ALIGN=BOTTOM ALT="index" SRC="icons/index_motif.gif"></A> <BR>
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<B> Next:</B> <A NAME=tex2html3130 HREF="node129.html"> Branch CutsPrincipal </A>
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<B>Up:</B> <A NAME=tex2html3128 HREF="node126.html"> Irrational and Transcendental </A>
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<B> Previous:</B> <A NAME=tex2html3122 HREF="node127.html"> Exponential and Logarithmic </A>
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<HR> <P>
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<H2><A NAME=SECTION001652000000000000000>12.5.2. Trigonometric and Related Functions</A></H2>
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<P>
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Some of the functions in this section, such as <tt>abs</tt>
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and <tt>signum</tt>, are apparently unrelated
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to trigonometric functions when considered as functions of
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real numbers only. The way in which they are extended to
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operate on complex numbers makes the trigonometric connection clear.
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<P>
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<BR><b>[Function]</b><BR>
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<tt>abs <i>number</i></tt><P>Returns the absolute value of the argument. For a non-complex number <i>x</i>,
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<P><pre>
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(abs <i>x</i>) == (if (minusp <i>x</i>) (- <i>x</i>) <i>x</i>)
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</pre><P>
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and the result is always of the same type as the argument.
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<P>
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For a complex number <i>z</i>, the absolute value may be computed as
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<P><pre>
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(sqrt (+ (expt (realpart <i>z</i>) 2) (expt (imagpart <i>z</i>) 2)))
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</pre><P>
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<P>
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<hr>
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<b>Implementation note:</b> The careful implementor will not use this formula directly
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for all complex numbers
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but will instead handle very large or very small components specially
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to avoid intermediate overflow or underflow.
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<hr>
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<P>
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For example:
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<P><pre>
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(abs #c(3.0 -4.0)) => 5.0
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</pre><P>
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The result of <tt>(abs #c(3 4))</tt> may be either <tt>5</tt> or <tt>5.0</tt>,
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depending on the implementation.
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<P>
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<BR><b>[Function]</b><BR>
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<tt>phase <i>number</i></tt><P>The phase of a number is the angle part of its polar representation
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as a complex number. That is,
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<P><pre>
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(phase <i>z</i>) == (atan (imagpart <i>z</i>) (realpart <i>z</i>))
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</pre><P>
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<img align=bottom alt="old_change_begin" src="gif/old_change_begin.gif"><br>
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The result is in radians, in the range <b>-pi</b> (exclusive)
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to <b>+pi</b> (inclusive). The phase of a positive non-complex number
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is zero; that of a negative non-complex number is <b>pi</b>.
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The phase of zero is arbitrarily defined to be zero.
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<br><img align=bottom alt="old_change_end" src="gif/old_change_end.gif">
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<P>
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<img align=bottom alt="change_begin" src="gif/change_begin.gif"><br>
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X3J13 voted in January 1989
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(IEEE-ATAN-BRANCH-CUT) <A NAME=11946> </A>
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to specify certain floating-point behavior when minus zero is supported;
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<tt>phase</tt> is still defined in terms of <tt>atan</tt> as above,
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but thanks to a change in <tt>atan</tt> the range of <tt>phase</tt>
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becomes <b>-pi</b> <i>inclusive</i> to <b>+pi</b> inclusive. The value <b>-</b>
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results from an argument
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whose real part is negative and whose imaginary
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part is minus zero. The <tt>phase</tt> function therefore has a branch cut
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along the negative real axis. The phase of +0+0<i>i</i> is +0, of +0-0<i>i</i> is -0,
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of -0+0<i>i</i> is +<b>pi</b>, and of -0-0<i>i</i> is -<b>pi</b>.
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<br><img align=bottom alt="change_end" src="gif/change_end.gif">
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<P>
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If the argument is a complex floating-point number, the result
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is a floating-point number of the same type as the components of
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the argument.
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If the argument is a floating-point number, the result is a
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floating-point number of the same type.
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If the argument is a rational number or complex rational number, the result
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is a single-format floating-point number.
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<P>
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<BR><b>[Function]</b><BR>
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<tt>signum <i>number</i></tt><P>By definition,
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<P><pre>
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(signum <i>x</i>) == (if (zerop <i>x</i>) <i>x</i> (/ <i>x</i> (abs <i>x</i>)))
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</pre><P>
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For a rational number, <tt>signum</tt> will return one of <tt>-1</tt>, <tt>0</tt>, or <tt>1</tt>
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according to whether the number is negative, zero, or positive.
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For a floating-point number, the result will be a floating-point number
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of the same format whose value is -1, 0, or 1.
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For a complex number <i>z</i>, <tt>(signum <i>z</i>)</tt> is a complex number of
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the same phase but with unit magnitude, unless <i>z</i> is a complex zero,
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in which case the result is <i>z</i>.
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For example:
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<P><pre>
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(signum 0) => 0
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(signum -3.7L5) => -1.0L0
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(signum 4/5) => 1
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(signum #C(7.5 10.0)) => #C(0.6 0.8)
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(signum #C(0.0 -14.7)) => #C(0.0 -1.0)
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</pre><P>
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For non-complex rational numbers, <tt>signum</tt> is a rational function,
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but it may be irrational for complex arguments.
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<P>
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<BR><b>[Function]</b><BR>
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<tt>sin <i>radians</i> <BR></tt><tt>cos <i>radians</i> <BR></tt><tt>tan <i>radians</i></tt><P><tt>sin</tt> returns the sine of the argument, <tt>cos</tt> the cosine,
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and <tt>tan</tt> the tangent. The argument is in radians.
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The argument may be complex.
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<P>
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<BR><b>[Function]</b><BR>
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<tt>cis <i>radians</i></tt><P>This computes <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40831.gif">.
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The name <tt>cis</tt> means ``cos + <i>i</i> sin,'' because
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<IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40832.gif">.
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The argument is in
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radians and may be any non-complex number. The result is a complex
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number whose real part is the cosine of the argument and whose imaginary
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part is the sine. Put another way, the result is a complex number whose
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phase is the equal to the argument (mod 2)
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and whose magnitude is unity.
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<P>
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<hr>
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<b>Implementation note:</b> Often it is cheaper to calculate the sine and cosine
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of a single angle together than to perform two disjoint calculations.
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<hr>
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<P>
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<BR><b>[Function]</b><BR>
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<tt>asin <i>number</i> <BR></tt><tt>acos <i>number</i></tt><P><tt>asin</tt> returns the arc sine of the argument, and <tt>acos</tt> the arc cosine.
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The result is in radians. The argument may be complex.
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<P>
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The arc sine and arc cosine functions may be defined mathematically for
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an argument <i>z</i> as follows:
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<PRE><TT>
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Arc sine <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40833.gif">
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Arc cosine <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40834.gif">
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<P>
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</TT></PRE>
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Note that the result of <tt>asin</tt> or <tt>acos</tt> may be
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complex even if the argument is not complex; this occurs
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when the absolute value of the argument is greater than 1.
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<P>
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<img align=bottom alt="change_begin" src="gif/change_begin.gif"><br>
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Kahan [<A HREF="node368.html#KAHANCOMPLEXFNS">25</A>] suggests for <tt>acos</tt> the
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defining formula
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<PRE><TT>
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Arc cosine <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40835.gif">
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<P>
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</TT></PRE>
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or even the much simpler <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40836.gif">. Both equations are mathematically
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equivalent to the formula shown above.
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<br><img align=bottom alt="change_end" src="gif/change_end.gif">
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<P>
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<hr>
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<b>Implementation note:</b> These formulae are mathematically correct, assuming
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completely accurate computation. They may be terrible methods for
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floating-point computation. Implementors should consult a good text on
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numerical analysis. The formulae given above are not necessarily
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the simplest ones for real-valued computations, either; they are chosen
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to define the branch cuts in desirable ways for the complex case.
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<hr>
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<P>
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<BR><b>[Function]</b><BR>
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<tt>atan <i>y</i> &optional <i>x</i></tt><P>An arc tangent is calculated and the result is returned in radians.
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<P>
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With two arguments <i>y</i> and <i>x</i>, neither argument may be complex.
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The result is the arc tangent of the quantity <i>y/x</i>.
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The signs of <i>y</i> and <i>x</i> are used to derive quadrant
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information; moreover, <i>x</i> may be zero provided
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<i>y</i> is not zero. The value of <tt>atan</tt> is always between
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<b>-pi</b> (exclusive) and <b>+pi</b> (inclusive).
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The following table details various special cases.
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<P>
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<P><IMG ALIGN=BOTTOM ALT="" SRC="_24769_tabular12036.gif"><P>
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<P>
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<img align=bottom alt="change_begin" src="gif/change_begin.gif"><br>
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X3J13 voted in January 1989
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(IEEE-ATAN-BRANCH-CUT) <A NAME=12073> </A>
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to specify certain floating-point behavior when minus zero is supported.
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When there is a minus zero, the preceding table must be modified slightly:
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<P>
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<P><IMG ALIGN=BOTTOM ALT="" SRC="_24769_tabular12075.gif"><P>
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<P>
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Note that the case <b><i>y</i>=0,<i>x</i>=0</b> is an error in the absence of minus zero,
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but the four cases <b><i>y</i>=0,<i>x</i>=0</b> are defined in the presence of minus zero.
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<br><img align=bottom alt="change_end" src="gif/change_end.gif">
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<P>
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<img align=bottom alt="old_change_begin" src="gif/old_change_begin.gif"><br>
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With only one argument <i>y</i>, the argument may be complex.
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The result is the arc tangent of <i>y</i>, which may be defined by
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the following formula:
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<PRE>
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Arc tangent <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40837.gif">
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</PRE>
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<br><img align=bottom alt="old_change_end" src="gif/old_change_end.gif">
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<P>
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<hr>
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<b>Implementation note:</b> This formula is mathematically correct, assuming
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completely accurate computation. It may be a terrible method for
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floating-point computation. Implementors should consult a good text on
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numerical analysis. The formula given above is not necessarily
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the simplest one for real-valued computations, either; it is chosen
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to define the branch cuts in desirable ways for the complex case.
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<hr>
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<P>
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<img align=bottom alt="change_begin" src="gif/change_begin.gif"><br>
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X3J13 voted in January 1989
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(COMPLEX-ATAN-BRANCH-CUT) <A NAME=12141> </A>
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to replace the preceding formula with the formula
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<PRE>
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log(1+<i>iy</i>) - log(1-<i>iy</i>)
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Arc tangent ---------------------
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2<i>i</i>
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</PRE>
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This change alters the direction of continuity for the
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branch cuts, which alters the result returned by <tt>atan</tt>
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only for arguments on the imaginary axis that
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are of magnitude greater than 1.
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See section <A HREF="node129.html#BRANCHCUTSSECTION">12.5.3</A> for further details.
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<br><img align=bottom alt="change_end" src="gif/change_end.gif">
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<P>
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For a non-complex argument <i>y</i>, the result is non-complex and lies between
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<IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap41199.gif"> and <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap41251.gif"> (both exclusive).
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<P>
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<hr>
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<b>Compatibility note:</b> MacLisp has a function called <tt>atan</tt> whose
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range is from 0 to <b>2</b>. Almost every other programming language
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(ANSI Fortran, IBM PL/1, Interlisp) has a two-argument arc tangent
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function with range <b>-pi</b> to <b>+pi</b>.
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Lisp Machine Lisp provides two two-argument
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arc tangent functions, <tt>atan</tt> (compatible with MacLisp)
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and <tt>atan2</tt> (compatible with all others).
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<P>
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Common Lisp makes two-argument <tt>atan</tt> the standard one
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with range <b>-pi</b> to <b>+pi</b>. Observe that this makes
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the one-argument and two-argument versions of <tt>atan</tt> compatible
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in the sense that the branch cuts do not fall in different places.
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The Interlisp one-argument function <tt>arctan</tt> has a range
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from 0 to <b>pi</b>, while nearly every other programming language
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provides the range <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap41199.gif"> to <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap41251.gif"> for
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one-argument arc tangent!
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Nevertheless, since Interlisp uses the standard two-argument
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version of arc tangent, its branch cuts are inconsistent anyway.
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<hr>
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<P>
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<BR><b>[Constant]</b><BR>
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<tt>pi</tt><P>This global variable has as its value the best possible approximation to
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<b>pi</b> in <i>long</i> floating-point format.
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For example:
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<P><pre>
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(defun sind (x) ;The argument is in degrees
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(sin (* x (/ (float pi x) 180))))
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</pre><P>
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An approximation to <b>pi</b> in some other precision can
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be obtained by writing <tt>(float pi <i>x</i>)</tt>, where <i>x</i> is a
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floating-point number of the desired precision,
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or by writing <tt>(coerce pi <i>type</i>)</tt>, where <i>type</i> is the
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name of the desired type, such as <tt>short-float</tt>.
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<P>
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<BR><b>[Function]</b><BR>
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<tt>sinh <i>number</i> <BR></tt><tt>cosh <i>number</i> <BR></tt><tt>tanh <i>number</i> <BR></tt><tt>asinh <i>number</i> <BR></tt><tt>acosh <i>number</i> <BR></tt><tt>atanh <i>number</i></tt>
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<P>
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<img align=bottom alt="old_change_begin" src="gif/old_change_begin.gif"><br>
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These functions compute the hyperbolic sine, cosine, tangent,
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arc sine, arc cosine, and arc tangent functions, which are mathematically
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defined for an argument <i>z</i> as follows:
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<PRE>
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Hyperbolic sine <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40839.gif">
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Hyperbolic cosine <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40840.gif">
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|
Hyperbolic tangent <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40841.gif">
|
||
|
|
Hyperbolic arc sine <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40842.gif">
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Hyperbolic arc cosine <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40843.gif">
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Hyperbolic arc tangent <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40844.gif"> <b>WRONG!</b>
|
||
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</PRE>
|
||
|
|
<img align=bottom alt="old_change_end" src="gif/old_change_end.gif">
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|
<P>
|
||
|
|
<img align=bottom alt="change_begin" src="gif/change_begin.gif"><br>
|
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|
|
WARNING! <i>The formula shown above for hyperbolic arc tangent is incorrect.</i>
|
||
|
|
It is not a matter of incorrect branch cuts; it simply does not compute anything
|
||
|
|
like a hyperbolic arc tangent. This unfortunate error in the first edition
|
||
|
|
was the result of mistranscribing a (correct) APL formula from Penfield's paper
|
||
|
|
[<A HREF="node368.html#APLBRANCHCUTS">36</A>]. The formula should have been transcribed as
|
||
|
|
<PRE><TT>
|
||
|
|
Hyperbolic arc tangent <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40845.gif">
|
||
|
|
<P>
|
||
|
|
</TT></PRE>
|
||
|
|
A proposal was submitted to X3J13 in September 1989 to replace the
|
||
|
|
formulae for <tt>acosh</tt> and <tt>atanh</tt>.
|
||
|
|
See section <A HREF="node129.html#BRANCHCUTSSECTION">12.5.3</A> for further discussion.
|
||
|
|
<br><img align=bottom alt="change_end" src="gif/change_end.gif">
|
||
|
|
<P>
|
||
|
|
Note that the result of <tt>acosh</tt> may be
|
||
|
|
complex even if the argument is not complex; this occurs
|
||
|
|
when the argument is less than 1.
|
||
|
|
Also, the result of <tt>atanh</tt> may be
|
||
|
|
complex even if the argument is not complex; this occurs
|
||
|
|
when the absolute value of the argument is greater than 1.
|
||
|
|
<P>
|
||
|
|
<hr>
|
||
|
|
<b>Implementation note:</b> These formulae are mathematically correct, assuming
|
||
|
|
completely accurate computation. They may be terrible methods for
|
||
|
|
floating-point computation. Implementors should consult a good text on
|
||
|
|
numerical analysis. The formulae given above are not necessarily
|
||
|
|
the simplest ones for real-valued computations, either; they are chosen
|
||
|
|
to define the branch cuts in desirable ways for the complex case.
|
||
|
|
<hr>
|
||
|
|
<P>
|
||
|
|
<BR> <HR><A NAME=tex2html3129 HREF="node129.html"><IMG ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME=tex2html3127 HREF="node126.html"><IMG ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME=tex2html3121 HREF="node127.html"><IMG ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <A NAME=tex2html3131 HREF="node1.html"><IMG ALIGN=BOTTOM ALT="contents" SRC="icons/contents_motif.gif"></A> <A NAME=tex2html3132 HREF="index.html"><IMG ALIGN=BOTTOM ALT="index" SRC="icons/index_motif.gif"></A> <BR>
|
||
|
|
<B> Next:</B> <A NAME=tex2html3130 HREF="node129.html"> Branch CutsPrincipal </A>
|
||
|
|
<B>Up:</B> <A NAME=tex2html3128 HREF="node126.html"> Irrational and Transcendental </A>
|
||
|
|
<B> Previous:</B> <A NAME=tex2html3122 HREF="node127.html"> Exponential and Logarithmic </A>
|
||
|
|
<HR> <P>
|
||
|
|
<HR>
|
||
|
|
<P><ADDRESS>
|
||
|
|
AI.Repository@cs.cmu.edu
|
||
|
|
</ADDRESS>
|
||
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