git-svn-id: https://lxsd.femto-st.fr/svn/fvn@39 b657c933-2333-4658-acf2-d3c7c2708721

daniau
1 parent e1aefab233
Showing 6 changed files with 2904 additions and 2062 deletions Side-by-side Diff
doc/besseli-mono.pdf
doc/besselj-mono.pdf
doc/besselk-mono.pdf
doc/bessely-mono.pdf
doc/fvn.pdf
doc/fvn.tex
@@ -300,7 +300,7 @@
 \begin{itemize}
  \item n (in) is an integer equal to the matrix rank
 \end{itemize}
-This function return the identity matrix of rank n, in the type of the left hand side. No generic interface for this one.
+This function return the identity matrix of rank n, in the specified type. No generic interface for this one.
  
  
  
@@ -585,7 +585,7 @@
  
  
 \section{Least square polynomial}
-fvn provide a function to find a least square polynomial of a given degree, for real in single or double precision. It is performed using Lapack subroutine sgelss (dgelss), which solve this problem using singular value decomposition.
+fvn provide a function to find a least square polynomial of a given degree, for real in single or double precision. It is performed using Lapack subroutine sgels (dgels), which solve this problem.
  
 \begin{verbatim}
 call fvn_lspoly(np,x,y,deg,coeff,status)
@@ -848,7 +848,7 @@
  
  
 \section{Special functions}
-Specials functions are available in fvn by using an implementation of fnlib \url{http://www.netlib.org/fn}. This can be used separatly from the rest of fvn by using the module \verb'fvn_fnlib' and linking the library \verb'libfvn_fnlib.a' . The module provides a generic interfaces to all the routines. Specific names of the routines are given in the description. The double complex versions of the routines are not present in the web version of fnlib, so these have been added, but not intensely tested.
+Specials functions are available in fvn by using an implementation of fnlib \url{http://www.netlib.org/fn} with some additions. This can be used separatly from the rest of fvn by using the module \verb'fvn_fnlib' and linking the library \verb'libfvn_fnlib.a' . The module provides a generic interfaces to all the routines. Specific names of the routines are given in the description.
  
 \paragraph{Important Note}
 Due to the addition of fnlib to fvn, some functions that were in fvn and are redondant will be removed from fvn, so update your code now and replace them with the fnlib version. These are listed here after :
  
@@ -1533,12 +1533,47 @@
  
 Specific interfaces : \verb'besj1,dbesj1'
  
+\subsubsection{bsjn}
+\begin{verbatim}
+ bsjn(n,x,factor,big)
+\end{verbatim}
+\begin{itemize}
+ \item n is an integer
+ \item x is a real
+ \item factor is an optional integer
+ \item big is an optional real
+\end{itemize}
+This function evaluates Bessel function of the first kind of order n (plotted in figure \ref{besseljnfamily}). These functions satisfy the recurrent relation \ref {bsjn}.
+\begin{equation}
+ \label{bsjn}
+J_{n+1}(x)={{2n}\over x} J_n(x) - J_{n-1}(x)
+\end{equation}
+This relation is directly used in upward direction to compute $J_n(x)$ for $x>n$. However it is unstable for $x<n$, therefore a Miller's Algorithm is used. The principle of this method is to use the reccurent relation downward from an arbitrary higher than n order with an arbitrary seed and then normalize the solution with \ref{bsjnnorm}
+\begin{equation}
+ \label{bsjnnorm}
+ 1=J_0+2J_2+2J_4+2J_6+...
+\end{equation}
+The optional parameters \verb'factor' and \verb'big' can be used to modify the behaviour of the algorithm. \verb'factor' is used in determining the arbitrary starting order ( an even integer near $n+\sqrt{factor~n}$), the default $factor$ value is 40. \verb'big' is a real determining the threshold for which anti-overflow counter measures has to be taken, default value is $1.10^{10}$
+
+By convenience, the routine accept $n=0$ and $n=1$, in that cases a call to $bsj0(x)$ or $bsj1(x)$ is actually performed.
+
+\begin{figure}
+ \begin{center}
+ \includegraphics[width=0.9\textwidth]{besselj-mono.pdf}
+ \caption{Bessel $J_n$ functions family}
+ \label{besseljnfamily}
+ \end{center}
+\end{figure}
+
+
+Specific interfaces : \verb'besjn,dbesjn'
+
 \subsubsection{bsy0}
 \begin{verbatim}
  bsy0(x)
 \end{verbatim}
 \begin{itemize}
- \item x is a real
+ \item x is a strictly positive real
 \end{itemize}
 This function evaluates the Bessel function of the second kind of order 0 defined by equation \ref{bsy0}
 \begin{equation}
@@ -1553,7 +1588,7 @@
  bsy1(x)
 \end{verbatim}
 \begin{itemize}
- \item x is a real
+ \item x is a strictly positive real
 \end{itemize}
 This function evaluates the Bessel function of the second kind of order 1 defined by equation \ref{bsy1}.
 \begin{equation}
@@ -1564,6 +1599,35 @@
  
 Specific interfaces : \verb'besy1,dbesy1'
  
+
+\subsubsection{bsyn}
+\begin{verbatim}
+ bsyn(n,x)
+\end{verbatim}
+\begin{itemize}
+ \item n is an integer
+ \item x is a strictly positive real
+\end{itemize}
+This function evaluates the Bessel function of the second kind of order n (plotted in figure \ref{besselynfamily}). These functions satisfy the recurrent relation \ref {bsyn}.
+\begin{equation}
+ \label{bsyn}
+Y_{n+1}(x)={{2n}\over x} Y_n(x) - Y_{n-1}(x)
+\end{equation}
+This recurrent relation is directly used in the upward direction to compute $Y_n(x)$.
+
+By convenience, the routine accept $n=0$ and $n=1$, in that cases a call to $bsy0(x)$ or $bsy1(x)$ is actually performed.
+
+\begin{figure}
+ \begin{center}
+ \includegraphics[width=0.9\textwidth]{bessely-mono.pdf}
+ \caption{Bessel $Y_n$ functions family}
+ \label{besselynfamily}
+ \end{center}
+\end{figure}
+
+
+Specific interfaces : \verb'besyn,dbesyn'
+
 \subsubsection{bsi0}
 \begin{verbatim}
  bsi0(x)
@@ -1594,6 +1658,42 @@
  
 Specific interfaces : \verb'besi1,dbesi1'
  
+\subsubsection{bsin}
+\begin{verbatim}
+ bsin(n,x,factor,big)
+\end{verbatim}
+\begin{itemize}
+ \item n is an integer
+ \item x is a real
+ \item factor is an optional integer
+ \item big is an optional real
+\end{itemize}
+This function evaluates Bessel function of the third kind of order n (plotted in figure \ref{besselinfamily}). These functions satisfy the recuurent relation \ref{bsin}
+\begin{equation}
+ \label{bsin}
+I_{n+1}(x)=-{{2n}\over x}I_n(x) + I_{n-1}(x)
+\end{equation}
+This relation is unstable in the upward direction, therefore a Miller's Algorithm is used to evaluate the function. Even if there's a usable normalization relation \ref{bsinnorm}, it is not used in the routine, instead normalization  is done by a simple call to \verb'bsi0(x)'.
+\begin{equation}
+ \label{bsinnorm}
+ 1=I_0-2I_2+2I_4-2I_6+....
+\end{equation}
+The optional parameters \verb'factor' and \verb'big' can be used to modify the behaviour of the algorithm. \verb'factor' is used in determining the arbitrary starting order ( an even integer near $n+\sqrt{factor~n}$), the default $factor$ value is 40. \verb'big' is a real determining the threshold for which anti-overflow counter measures has to be taken, default value is $1.10^{10}$
+
+By convenience, the routine accept $n=0$ and $n=1$, in that cases a call to $bsi0(x)$ or $bsi1(x)$ is actually performed.
+
+\begin{figure}
+ \begin{center}
+ \includegraphics[width=0.9\textwidth]{besseli-mono.pdf}
+ \caption{Bessel $I_n$ functions family}
+ \label{besselinfamily}
+ \end{center}
+\end{figure}
+
+
+Specific interfaces : \verb'besin,dbesin'
+
+
 \subsubsection{bsk0}
 \begin{verbatim}
  bsk0(x)
@@ -1624,6 +1724,36 @@
  
 Specific interfaces : \verb'besk1,dbesk1'
  
+\subsubsection{bskn}
+\begin{verbatim}
+ bskn(n,x)
+\end{verbatim}
+\begin{itemize}
+ \item n is an integer
+ \item x is strictly positive real
+\end{itemize}
+This function evaluates the modified Bessel function of the second kind of order n (plotted in figure \ref{besselknfamily}). These functions satisfy the recurrent relation \ref{bskn}
+
+\begin{equation}
+ \label{bskn}
+K_{n+1}(x)={{2n}\over x}K_n(x)+K_{n-1}(x)
+\end{equation}
+This recurrent relation is directly used in the upward direction to compute $K_n(x)$.
+
+By convenience, the routine accept $n=0$ and $n=1$, in that cases a call to $bsk0(x)$ or $bsk1(x)$ is actually performed.
+
+\begin{figure}
+ \begin{center}
+ \includegraphics[width=0.9\textwidth]{besselk-mono.pdf}
+ \caption{Bessel $K_n$ functions family}
+ \label{besselknfamily}
+ \end{center}
+\end{figure}
+
+
+Specific interface : \verb'beskn,dbeskn'
+
+
 \subsubsection{bsi0e}
 \begin{verbatim}
  bsi0e(x)
  
  
@@ -1765,17 +1895,17 @@
 \end{itemize}
  
 This function evaluates the exponentially scaled Airy function defined in equation \ref{aie}.
-\begin{equation}
- \label{aie}
-aie(x)=Ai(x) \textrm{~if~}x\leq0 \qquad \qquad aie(x)=e^{{2\over3}x^{3\over2}}Ai(x)\rm{~if~}x>0
-\end{equation}
 %\begin{equation}
-%  \left{\begin{array}{ll}
-%  Ai(x) & \textrm{~if~}x\leq0 \\
-%  e^{{2\over3}x^{3\over2}}Ai(x) & \textrm{~if~}x>0
-%\end{array}\right. 
-%\label{aie}
+% \label{aie}
+%aie(x)=Ai(x) \textrm{~if~}x\leq0 \qquad \qquad aie(x)=e^{{2\over3}x^{3\over2}}Ai(x)\rm{~if~}x>0
 %\end{equation}
+\begin{equation}
+\label{aie}
+  aie(x)=\left\{\begin{array}{ll}
+  Ai(x) & \textrm{~if~}x\leq0 \\
+  e^{{2\over3}x^{3\over2}}Ai(x) & \textrm{~if~}x>0
+\end{array}\right.
+\end{equation}
  
 Specific interfaces : \verb'aie,daie'
  
  
@@ -1788,9 +1918,16 @@
  \item x is a real
 \end{itemize}
 This function evaluates the exponentially scaled Airy function of the second kind defined in equation \ref{bie}.
+%\begin{equation}
+% \label{bie}
+%bie(x)=Bi(x)\textrm{~if~}x\leq0 \qquad \qquad bie(x)=e^{-{2\over3}x^{3\over2}}Bi(x)\rm{~if~}x>0
+%\end{equation}
 \begin{equation}
  \label{bie}
-bie(x)=Bi(x)\textrm{~if~}x\leq0 \qquad \qquad bie(x)=e^{-{2\over3}x^{3\over2}}Bi(x)\rm{~if~}x>0
+bie(x)=\left\{\begin{array}{ll}
+Bi(x) & \textrm{~if~}x\leq0 \\
+e^{-{2\over3}x^{3\over2}}Bi(x) & \textrm{~if~}x>0
+\end{array}\right.
 \end{equation}
  
 Specific interfaces : \verb'bie,dbie'
  
  
@@ -1803,11 +1940,19 @@
  \item x is a real
 \end{itemize}
 This function evaluates the exponentially scaled derivative of the Airy function as defined in equation \ref{aide}.
+%\begin{equation}
+% \label{aide}
+%aie(x)=Ai^\prime(x) \textrm{~if~}x\leq0 \qquad \qquad aie(x)=e^{{2\over3}x^{3\over2}}Ai^\prime(x)\rm{~if~}x>0
+%\end{equation}
 \begin{equation}
  \label{aide}
-aie(x)=Ai^\prime(x) \textrm{~if~}x\leq0 \qquad \qquad aie(x)=e^{{2\over3}x^{3\over2}}Ai^\prime(x)\rm{~if~}x>0
+aide(x)=\left\{\begin{array}{ll}
+Ai^\prime(x) & \textrm{~if~}x\leq0 \\
+e^{{2\over3}x^{3\over2}}Ai^\prime(x) & \textrm{~if~}x>0
+\end{array}\right.
 \end{equation}
  
+
 Specific interfaces : \verb'aide,daide'
  
  
@@ -1821,7 +1966,10 @@
 This function evaluates the exponentially scaled derivative of the Airy function of the second kind as defined in equation \ref{bide}.
 \begin{equation}
  \label{bide}
-bie(x)=Bi^\prime(x)\textrm{~if~}x\leq0 \qquad \qquad bie(x)=e^{-{2\over3}x^{3\over2}}Bi^\prime(x)\rm{~if~}x>0
+bie(x)=\left\{\begin{array}{ll}
+Bi^\prime(x) & \textrm{~if~}x\leq0 \\
+e^{-{2\over3}x^{3\over2}}Bi^\prime(x) & \textrm{~if~}x>0
+\end{array}\right.
 \end{equation}
  
 Specific interfaces : \verb'bide,dbide'
...	...	@@ -300,7 +300,7 @@
300	300	\begin{itemize}
301	301	\item n (in) is an integer equal to the matrix rank
302	302	\end{itemize}
303		-This function return the identity matrix of rank n, in the type of the left hand side. No generic interface for this one.
	303	+This function return the identity matrix of rank n, in the specified type. No generic interface for this one.
304	304
305	305
306	306
...	...	@@ -585,7 +585,7 @@
585	585
586	586
587	587	\section{Least square polynomial}
588		-fvn provide a function to find a least square polynomial of a given degree, for real in single or double precision. It is performed using Lapack subroutine sgelss (dgelss), which solve this problem using singular value decomposition.
	588	+fvn provide a function to find a least square polynomial of a given degree, for real in single or double precision. It is performed using Lapack subroutine sgels (dgels), which solve this problem.
589	589
590	590	\begin{verbatim}
591	591	call fvn_lspoly(np,x,y,deg,coeff,status)
...	...	@@ -848,7 +848,7 @@
848	848
849	849
850	850	\section{Special functions}
851		-Specials functions are available in fvn by using an implementation of fnlib \url{http://www.netlib.org/fn}. This can be used separatly from the rest of fvn by using the module \verb'fvn_fnlib' and linking the library \verb'libfvn_fnlib.a' . The module provides a generic interfaces to all the routines. Specific names of the routines are given in the description. The double complex versions of the routines are not present in the web version of fnlib, so these have been added, but not intensely tested.
	851	+Specials functions are available in fvn by using an implementation of fnlib \url{http://www.netlib.org/fn} with some additions. This can be used separatly from the rest of fvn by using the module \verb'fvn_fnlib' and linking the library \verb'libfvn_fnlib.a' . The module provides a generic interfaces to all the routines. Specific names of the routines are given in the description.
852	852
853	853	\paragraph{Important Note}
854	854	Due to the addition of fnlib to fvn, some functions that were in fvn and are redondant will be removed from fvn, so update your code now and replace them with the fnlib version. These are listed here after :
855	855
...	...	@@ -1533,12 +1533,47 @@
1533	1533
1534	1534	Specific interfaces : \verb'besj1,dbesj1'
1535	1535
	1536	+\subsubsection{bsjn}
	1537	+\begin{verbatim}
	1538	+ bsjn(n,x,factor,big)
	1539	+\end{verbatim}
	1540	+\begin{itemize}
	1541	+ \item n is an integer
	1542	+ \item x is a real
	1543	+ \item factor is an optional integer
	1544	+ \item big is an optional real
	1545	+\end{itemize}
	1546	+This function evaluates Bessel function of the first kind of order n (plotted in figure \ref{besseljnfamily}). These functions satisfy the recurrent relation \ref {bsjn}.
	1547	+\begin{equation}
	1548	+ \label{bsjn}
	1549	+J_{n+1}(x)={{2n}\over x} J_n(x) - J_{n-1}(x)
	1550	+\end{equation}
	1551	+This relation is directly used in upward direction to compute $J_n(x)$ for $x>n$. However it is unstable for $x<n$, therefore a Miller's Algorithm is used. The principle of this method is to use the reccurent relation downward from an arbitrary higher than n order with an arbitrary seed and then normalize the solution with \ref{bsjnnorm}
	1552	+\begin{equation}
	1553	+ \label{bsjnnorm}
	1554	+ 1=J_0+2J_2+2J_4+2J_6+...
	1555	+\end{equation}
	1556	+The optional parameters \verb'factor' and \verb'big' can be used to modify the behaviour of the algorithm. \verb'factor' is used in determining the arbitrary starting order ( an even integer near $n+\sqrt{factor~n}$), the default $factor$ value is 40. \verb'big' is a real determining the threshold for which anti-overflow counter measures has to be taken, default value is $1.10^{10}$
	1557	+
	1558	+By convenience, the routine accept $n=0$ and $n=1$, in that cases a call to $bsj0(x)$ or $bsj1(x)$ is actually performed.
	1559	+
	1560	+\begin{figure}
	1561	+ \begin{center}
	1562	+ \includegraphics[width=0.9\textwidth]{besselj-mono.pdf}
	1563	+ \caption{Bessel $J_n$ functions family}
	1564	+ \label{besseljnfamily}
	1565	+ \end{center}
	1566	+\end{figure}
	1567	+
	1568	+
	1569	+Specific interfaces : \verb'besjn,dbesjn'
	1570	+
1536	1571	\subsubsection{bsy0}
1537	1572	\begin{verbatim}
1538	1573	bsy0(x)
1539	1574	\end{verbatim}
1540	1575	\begin{itemize}
1541		- \item x is a real
	1576	+ \item x is a strictly positive real
1542	1577	\end{itemize}
1543	1578	This function evaluates the Bessel function of the second kind of order 0 defined by equation \ref{bsy0}
1544	1579	\begin{equation}
...	...	@@ -1553,7 +1588,7 @@
1553	1588	bsy1(x)
1554	1589	\end{verbatim}
1555	1590	\begin{itemize}
1556		- \item x is a real
	1591	+ \item x is a strictly positive real
1557	1592	\end{itemize}
1558	1593	This function evaluates the Bessel function of the second kind of order 1 defined by equation \ref{bsy1}.
1559	1594	\begin{equation}
...	...	@@ -1564,6 +1599,35 @@
1564	1599
1565	1600	Specific interfaces : \verb'besy1,dbesy1'
1566	1601
	1602	+
	1603	+\subsubsection{bsyn}
	1604	+\begin{verbatim}
	1605	+ bsyn(n,x)
	1606	+\end{verbatim}
	1607	+\begin{itemize}
	1608	+ \item n is an integer
	1609	+ \item x is a strictly positive real
	1610	+\end{itemize}
	1611	+This function evaluates the Bessel function of the second kind of order n (plotted in figure \ref{besselynfamily}). These functions satisfy the recurrent relation \ref {bsyn}.
	1612	+\begin{equation}
	1613	+ \label{bsyn}
	1614	+Y_{n+1}(x)={{2n}\over x} Y_n(x) - Y_{n-1}(x)
	1615	+\end{equation}
	1616	+This recurrent relation is directly used in the upward direction to compute $Y_n(x)$.
	1617	+
	1618	+By convenience, the routine accept $n=0$ and $n=1$, in that cases a call to $bsy0(x)$ or $bsy1(x)$ is actually performed.
	1619	+
	1620	+\begin{figure}
	1621	+ \begin{center}
	1622	+ \includegraphics[width=0.9\textwidth]{bessely-mono.pdf}
	1623	+ \caption{Bessel $Y_n$ functions family}
	1624	+ \label{besselynfamily}
	1625	+ \end{center}
	1626	+\end{figure}
	1627	+
	1628	+
	1629	+Specific interfaces : \verb'besyn,dbesyn'
	1630	+
1567	1631	\subsubsection{bsi0}
1568	1632	\begin{verbatim}
1569	1633	bsi0(x)
...	...	@@ -1594,6 +1658,42 @@
1594	1658
1595	1659	Specific interfaces : \verb'besi1,dbesi1'
1596	1660
	1661	+\subsubsection{bsin}
	1662	+\begin{verbatim}
	1663	+ bsin(n,x,factor,big)
	1664	+\end{verbatim}
	1665	+\begin{itemize}
	1666	+ \item n is an integer
	1667	+ \item x is a real
	1668	+ \item factor is an optional integer
	1669	+ \item big is an optional real
	1670	+\end{itemize}
	1671	+This function evaluates Bessel function of the third kind of order n (plotted in figure \ref{besselinfamily}). These functions satisfy the recuurent relation \ref{bsin}
	1672	+\begin{equation}
	1673	+ \label{bsin}
	1674	+I_{n+1}(x)=-{{2n}\over x}I_n(x) + I_{n-1}(x)
	1675	+\end{equation}
	1676	+This relation is unstable in the upward direction, therefore a Miller's Algorithm is used to evaluate the function. Even if there's a usable normalization relation \ref{bsinnorm}, it is not used in the routine, instead normalization is done by a simple call to \verb'bsi0(x)'.
	1677	+\begin{equation}
	1678	+ \label{bsinnorm}
	1679	+ 1=I_0-2I_2+2I_4-2I_6+....
	1680	+\end{equation}
	1681	+The optional parameters \verb'factor' and \verb'big' can be used to modify the behaviour of the algorithm. \verb'factor' is used in determining the arbitrary starting order ( an even integer near $n+\sqrt{factor~n}$), the default $factor$ value is 40. \verb'big' is a real determining the threshold for which anti-overflow counter measures has to be taken, default value is $1.10^{10}$
	1682	+
	1683	+By convenience, the routine accept $n=0$ and $n=1$, in that cases a call to $bsi0(x)$ or $bsi1(x)$ is actually performed.
	1684	+
	1685	+\begin{figure}
	1686	+ \begin{center}
	1687	+ \includegraphics[width=0.9\textwidth]{besseli-mono.pdf}
	1688	+ \caption{Bessel $I_n$ functions family}
	1689	+ \label{besselinfamily}
	1690	+ \end{center}
	1691	+\end{figure}
	1692	+
	1693	+
	1694	+Specific interfaces : \verb'besin,dbesin'
	1695	+
	1696	+
1597	1697	\subsubsection{bsk0}
1598	1698	\begin{verbatim}
1599	1699	bsk0(x)
...	...	@@ -1624,6 +1724,36 @@
1624	1724
1625	1725	Specific interfaces : \verb'besk1,dbesk1'
1626	1726
	1727	+\subsubsection{bskn}
	1728	+\begin{verbatim}
	1729	+ bskn(n,x)
	1730	+\end{verbatim}
	1731	+\begin{itemize}
	1732	+ \item n is an integer
	1733	+ \item x is strictly positive real
	1734	+\end{itemize}
	1735	+This function evaluates the modified Bessel function of the second kind of order n (plotted in figure \ref{besselknfamily}). These functions satisfy the recurrent relation \ref{bskn}
	1736	+
	1737	+\begin{equation}
	1738	+ \label{bskn}
	1739	+K_{n+1}(x)={{2n}\over x}K_n(x)+K_{n-1}(x)
	1740	+\end{equation}
	1741	+This recurrent relation is directly used in the upward direction to compute $K_n(x)$.
	1742	+
	1743	+By convenience, the routine accept $n=0$ and $n=1$, in that cases a call to $bsk0(x)$ or $bsk1(x)$ is actually performed.
	1744	+
	1745	+\begin{figure}
	1746	+ \begin{center}
	1747	+ \includegraphics[width=0.9\textwidth]{besselk-mono.pdf}
	1748	+ \caption{Bessel $K_n$ functions family}
	1749	+ \label{besselknfamily}
	1750	+ \end{center}
	1751	+\end{figure}
	1752	+
	1753	+
	1754	+Specific interface : \verb'beskn,dbeskn'
	1755	+
	1756	+
1627	1757	\subsubsection{bsi0e}
1628	1758	\begin{verbatim}
1629	1759	bsi0e(x)
1630	1760
1631	1761
...	...	@@ -1765,17 +1895,17 @@
1765	1895	\end{itemize}
1766	1896
1767	1897	This function evaluates the exponentially scaled Airy function defined in equation \ref{aie}.
1768		-\begin{equation}
1769		- \label{aie}
1770		-aie(x)=Ai(x) \textrm{~if~}x\leq0 \qquad \qquad aie(x)=e^{{2\over3}x^{3\over2}}Ai(x)\rm{~if~}x>0
1771		-\end{equation}
1772	1898	%\begin{equation}
1773		-% \left{\begin{array}{ll}
1774		-% Ai(x) & \textrm{~if~}x\leq0 \\
1775		-% e^{{2\over3}x^{3\over2}}Ai(x) & \textrm{~if~}x>0
1776		-%\end{array}\right.
1777		-%\label{aie}
	1899	+% \label{aie}
	1900	+%aie(x)=Ai(x) \textrm{~if~}x\leq0 \qquad \qquad aie(x)=e^{{2\over3}x^{3\over2}}Ai(x)\rm{~if~}x>0
1778	1901	%\end{equation}
	1902	+\begin{equation}
	1903	+\label{aie}
	1904	+ aie(x)=\left\{\begin{array}{ll}
	1905	+ Ai(x) & \textrm{~if~}x\leq0 \\
	1906	+ e^{{2\over3}x^{3\over2}}Ai(x) & \textrm{~if~}x>0
	1907	+\end{array}\right.
	1908	+\end{equation}
1779	1909
1780	1910	Specific interfaces : \verb'aie,daie'
1781	1911
1782	1912
...	...	@@ -1788,9 +1918,16 @@
1788	1918	\item x is a real
1789	1919	\end{itemize}
1790	1920	This function evaluates the exponentially scaled Airy function of the second kind defined in equation \ref{bie}.
	1921	+%\begin{equation}
	1922	+% \label{bie}
	1923	+%bie(x)=Bi(x)\textrm{~if~}x\leq0 \qquad \qquad bie(x)=e^{-{2\over3}x^{3\over2}}Bi(x)\rm{~if~}x>0
	1924	+%\end{equation}
1791	1925	\begin{equation}
1792	1926	\label{bie}
1793		-bie(x)=Bi(x)\textrm{~if~}x\leq0 \qquad \qquad bie(x)=e^{-{2\over3}x^{3\over2}}Bi(x)\rm{~if~}x>0
	1927	+bie(x)=\left\{\begin{array}{ll}
	1928	+Bi(x) & \textrm{~if~}x\leq0 \\
	1929	+e^{-{2\over3}x^{3\over2}}Bi(x) & \textrm{~if~}x>0
	1930	+\end{array}\right.
1794	1931	\end{equation}
1795	1932
1796	1933	Specific interfaces : \verb'bie,dbie'
1797	1934
1798	1935
...	...	@@ -1803,11 +1940,19 @@
1803	1940	\item x is a real
1804	1941	\end{itemize}
1805	1942	This function evaluates the exponentially scaled derivative of the Airy function as defined in equation \ref{aide}.
	1943	+%\begin{equation}
	1944	+% \label{aide}
	1945	+%aie(x)=Ai^\prime(x) \textrm{~if~}x\leq0 \qquad \qquad aie(x)=e^{{2\over3}x^{3\over2}}Ai^\prime(x)\rm{~if~}x>0
	1946	+%\end{equation}
1806	1947	\begin{equation}
1807	1948	\label{aide}
1808		-aie(x)=Ai^\prime(x) \textrm{~if~}x\leq0 \qquad \qquad aie(x)=e^{{2\over3}x^{3\over2}}Ai^\prime(x)\rm{~if~}x>0
	1949	+aide(x)=\left\{\begin{array}{ll}
	1950	+Ai^\prime(x) & \textrm{~if~}x\leq0 \\
	1951	+e^{{2\over3}x^{3\over2}}Ai^\prime(x) & \textrm{~if~}x>0
	1952	+\end{array}\right.
1809	1953	\end{equation}
1810	1954
	1955	+
1811	1956	Specific interfaces : \verb'aide,daide'
1812	1957
1813	1958
...	...	@@ -1821,7 +1966,10 @@
1821	1966	This function evaluates the exponentially scaled derivative of the Airy function of the second kind as defined in equation \ref{bide}.
1822	1967	\begin{equation}
1823	1968	\label{bide}
1824		-bie(x)=Bi^\prime(x)\textrm{~if~}x\leq0 \qquad \qquad bie(x)=e^{-{2\over3}x^{3\over2}}Bi^\prime(x)\rm{~if~}x>0
	1969	+bie(x)=\left\{\begin{array}{ll}
	1970	+Bi^\prime(x) & \textrm{~if~}x\leq0 \\
	1971	+e^{-{2\over3}x^{3\over2}}Bi^\prime(x) & \textrm{~if~}x>0
	1972	+\end{array}\right.
1825	1973	\end{equation}
1826	1974
1827	1975	Specific interfaces : \verb'bide,dbide'