Update of /project/cl-gsl/cvsroot/cl-gsl/doc In directory common-lisp.net:/tmp/cvs-serv6348 Modified Files: sf.tex Log Message: Convert more of the math to images. Date: Thu Mar 24 04:48:43 2005 Author: edenny Index: cl-gsl/doc/sf.tex diff -u cl-gsl/doc/sf.tex:1.1 cl-gsl/doc/sf.tex:1.2 --- cl-gsl/doc/sf.tex:1.1 Mon Mar 21 04:44:31 2005 +++ cl-gsl/doc/sf.tex Thu Mar 24 04:48:41 2005 @@ -7,16 +7,17 @@ \Css{.monobold { font-weight: bold; font-family: monospace;}} \Css{.floatright { float: right;}} +\Css{.floatleft { float: left;}} +\Css{.floatclear { clear: both;}} \def\monobf#1{\ifHtml \HCode{<span class="monobold">} #1 \else \textbf{\texttt{#1}} \fi \ifHtml \HCode{</span>} \fi} \def\func#1#2{\ifHtml \HCode{<span class="floatright">} \fi - {[}Function{]} \ifHtml \HCode{</span>} \fi - \monobf{#1} \texttt{#2}} - -\setcounter{section}{6} + {[}Function{]} \ifHtml \HCode{</span><span class="floatleft">} \fi + \monobf{#1} \texttt{#2} + \ifHtml \HCode{</span> <div class="floatclear"> </div>} \fi} \section{Special Functions} @@ -81,12 +82,12 @@ The following precision levels are available for the mode argument, \begin{description} -\item [GSL\_PREC\_DOUBLE~]Double-precision, a relative accuracy of approximately - 2 {*} 10\^{}-16. -\item [GSL\_PREC\_SINGLE~]Single-precision, a relative accuracy of approximately - 10\^{}-7. -\item [GSL\_PREC\_APPROX~]Approximate values, a relative accuracy of -approximately 5 {*} 10\^{}-4. +\item [+prec-double+] Double-precision, a relative accuracy of approximately + \(2 * 10^{-16}\). +\item [+prec-single+] Single-precision, a relative accuracy of approximately + \(10^{-1}\). +\item [+prec-approx+] Approximate values, a relative accuracy of +approximately \(5 * 10^{-4}\). \end{description} The approximate mode provides the fastest evaluation at the lowest accuracy. @@ -94,47 +95,47 @@ \subsection{Airy Functions and Derivatives} -The Airy functions Ai(x) and Bi(x) are defined by the integral representations, -For further information see Abramowitz \& Stegun, Section 10.4. -The \HCode{<span class="mystyle">} Airy functions \HCode{</span>} are defined in the header file gsl\_sf\_airy.h. +The Airy functions \(Ai(x)\) and \(Bi(x)\) are defined by the integral +representations, -\subsubsection{Airy Functions} +$$Ai(x) = \frac{1}{\pi} \int_0^\infty \cos(\frac{1}{3} t^3 + xt) dt$$ + +$$Bi(x) = \frac{1}{\pi} \int_0^\infty (e^{-t^3/3} + \sin(\frac{1}{3} t^3 + xt)) dt$$ +for further information see Abramowitz \& Stegun, Section 10.4. + +\subsubsection{Airy Functions} \func{airy-ai}{x mode => result} \func{airy-ai-e}{x mode => result, error, status} -These routines compute the Airy function Ai(x) with an accuracy +These routines compute the Airy function \(Ai(x)\) with an accuracy specified by mode. \func{airy-bi}{x mode => result} \func{airy-bi-e}{x mode => result, error, status} -These routines compute the Airy function Bi(x) with an accuracy +These routines compute the Airy function \(Bi(x)\) with an accuracy specified by mode. \func{airy-ai-scaled}{ x mode => result} - \func{airy-ai-scaled-e}{ x mode => result, error, status} - These routines compute a scaled version of the Airy function -S\_A(x) Ai(x). For x>0 the scaling factor S\_A(x) is $\backslash$exp(+(2/3) -x\^{}(3/2)), and is 1 for x<0. +\(S_A(x) Ai(x)\). For \(x>0\) the scaling factor \(S_A(x)\) is +\(e^{+(2/3)x^{3/2}}\), and is 1 for \(x<0\). \func{airy-bi-scaled}{ x mode => result} - \func{airy-bi-scaled-e}{ x mode => result, error, status} - These routines compute a scaled version of the Airy function -S\_B(x) Bi(x). For x>0 the scaling factor S\_B(x) is exp(-(2/3) x\^{}(3/2)), -and is 1 for x<0. +\(S_B(x) Bi(x)\). For \(x>0\) the scaling factor \(S_B(x)\) is +\(e^{-(2/3) x^{3/2}}\), and is 1 for \(x<0\). \subsubsection{Derivatives of Airy Functions} @@ -145,7 +146,7 @@ \func{airy-ai-deriv-e}{ x mode => result, error, status} -These routines compute the Airy function derivative Ai\verb|'|(x) +These routines compute the Airy function derivative \(Ai'(x)\) with an accuracy specified by mode. \func{airy-bi-deriv}{ x mode => result} @@ -154,7 +155,7 @@ \func{airy-bi-deriv-e}{ x mode => result, error, status} -These routines compute the Airy function derivative Bi\verb|'|(x) +These routines compute the Airy function derivative \(Bi'(x)\) with an accuracy specified by mode. \func{airy-ai-deriv-scaled}{ x mode => result} @@ -163,7 +164,7 @@ These routines compute the derivative of the scaled Airy function -S\_A(x) Ai(x). +\(S_A(x) Ai(x)\). \func{airy-bi-deriv-scaled}{ x mode => result} @@ -172,7 +173,7 @@ These routines compute the derivative of the scaled Airy function -S\_B(x) Bi(x). +\(S_B(x) Bi(x)\). \subsubsection{Zeros of Airy Functions} @@ -184,7 +185,7 @@ These routines compute the location of the s-th zero of the -Airy function Ai(x). +Airy function \(Ai(x)\). \func{airy-zero-bi}{ s => result} @@ -193,7 +194,7 @@ These routines compute the location of the s-th zero of the -Airy function Bi(x). +Airy function \(Bi(x)\). \subsubsection{Zeros of Derivatives of Airy Functions} @@ -205,7 +206,7 @@ These routines compute the location of the s-th zero of the -Airy function derivative Ai\verb|'|(x). +Airy function derivative \(Ai'(x)\). \func{airy-zero-bi-deriv}{ s => result} @@ -214,7 +215,7 @@ These routines compute the location of the s-th zero of the -Airy function derivative Bi\verb|'|(x). +Airy function derivative \(Bi'(x)\). \subsection{Bessel Functions} @@ -260,7 +261,8 @@ This routine computes the values of the regular cylindrical -Bessel functions \(J_n(x)\) for n from nmin to nmax inclusive, storing +Bessel functions \(J_n(x)\) for \texttt{n} from \texttt{nmin} to +\texttt{nmax} inclusive, storing the results in the array result-array. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values. @@ -299,7 +301,8 @@ This routine computes the values of the irregular cylindrical -Bessel functions \(Y_n(x)\) for n from nmin to nmax inclusive, storing +Bessel functions \(Y_n(x)\) for \texttt{n} from \texttt{nmin} +to \texttt{nmax} inclusive, storing the results in the array result-array. The domain of the function is \(x>0\). The values are computed using recurrence relations, for efficiency, @@ -339,7 +342,8 @@ This routine computes the values of the regular modified cylindrical -Bessel functions \(I_n(x)\) for n from nmin to nmax inclusive, storing +Bessel functions \(I_n(x)\) for \texttt{n} from \texttt{nmin} to +\texttt{nmax} inclusive, storing the results in the array result-array. The start of the range nmin must be positive or zero. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from @@ -376,8 +380,8 @@ This routine computes the values of the scaled regular cylindrical -Bessel functions \(e^{-|x|} I_n(x)\) for n from nmin to -nmax inclusive, storing the results in the array result-array. The +Bessel functions \(e^{-|x|} I_n(x)\) for \texttt{n} from \texttt{nmin} to +\texttt{nmax} inclusive, storing the results in the array result-array. The start of the range nmin must be positive or zero. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.