Merge pull request #589 from tkoeppe/imagine

[complex] Clarify the return value range of log(x)

Author: zygoloid

source/numerics.tex

@@ -307,7 +307,7 @@
   template<class T> T norm(const complex<T>&);
 
   template<class T> complex<T> conj(const complex<T>&);
-  template <class T> complex<T> proj(const complex<T>&);
+  template<class T> complex<T> proj(const complex<T>&);
   template<class T> complex<T> polar(const T&, const T& = 0);
 
   // \ref{complex.transcendentals}, transcendentals:
@@ -325,9 +325,9 @@
   template<class T> complex<T> log  (const complex<T>&);
   template<class T> complex<T> log10(const complex<T>&);
 
-  template<class T> complex<T> pow(const complex<T>&, const T&);
-  template<class T> complex<T> pow(const complex<T>&, const complex<T>&);
-  template<class T> complex<T> pow(const T&, const complex<T>&);
+  template<class T> complex<T> pow  (const complex<T>&, const T&);
+  template<class T> complex<T> pow  (const complex<T>&, const complex<T>&);
+  template<class T> complex<T> pow  (const T&, const complex<T>&);
 
   template<class T> complex<T> sin  (const complex<T>&);
   template<class T> complex<T> sinh (const complex<T>&);
@@ -1119,7 +1119,7 @@
 \begin{itemdescr}
 \pnum
 \returns
-The complex base e exponential of \tcode{x}.
+The complex base-$e$ exponential of \tcode{x}.
 \end{itemdescr}
 
 \indexlibrary{\idxcode{log}!\idxcode{complex}}%
@@ -1130,18 +1130,13 @@
 \begin{itemdescr}
 \pnum
 \notes
-the branch cuts are along the negative real axis.
+The branch cuts are along the negative real axis.
 
 \pnum
 \returns
-The complex natural (base e) logarithm of \tcode{x},
-in the range of a strip mathematically unbounded along the
-real axis and in the interval \crange{-i times pi}{i times pi}
-along the imaginary axis.
-When \tcode{x} is a negative real
-number,
-\tcode{imag(log(x))}
-is pi.
+The complex natural (base-$e$) logarithm of \tcode{x}. For all \tcode{x},
+\tcode{imag(log(x))} lies in the interval \crange{$-\pi$}{$\pi$}, and
+when \tcode{x} is a negative real number, \tcode{imag(log(x))} is $\pi$.
 \end{itemdescr}
 
 \indexlibrary{\idxcode{log10}!\idxcode{complex}}%
@@ -1152,40 +1147,39 @@
 \begin{itemdescr}
 \pnum
 \notes
-the branch cuts are along the negative real axis.
+The branch cuts are along the negative real axis.
 
 \pnum
 \returns
-The complex common (base 10) logarithm of \tcode{x}, defined as
-\tcode{log(x)/log(10)}.
+The complex common (base-$10$) logarithm of \tcode{x}, defined as
+\tcode{log(x) / log(10)}.
 \end{itemdescr}
 
 \indexlibrary{\idxcode{pow}!\idxcode{complex}}%
 \begin{itemdecl}
-template<class T>
-  complex<T> pow(const complex<T>& x, const complex<T>& y);
-template<class T> complex<T> pow  (const complex<T>& x, const T& y);
-template<class T> complex<T> pow  (const T& x, const complex<T>& y);
+template<class T> complex<T> pow(const complex<T>& x, const complex<T>& y);
+template<class T> complex<T> pow(const complex<T>& x, const T& y);
+template<class T> complex<T> pow(const T& x, const complex<T>& y);
 \end{itemdecl}
 
 \begin{itemdescr}
 \pnum
 \notes
-the branch cuts are along the negative real axis.
+The branch cuts are along the negative real axis.
 
 \pnum
 \returns
-The complex power of base \tcode{x} raised to the \tcode{y}-th power,
+The complex power of base \tcode{x} raised to the \tcode{y}$^\text{th}$ power,
 defined as
-\tcode{exp(y*log(x))}.
+\tcode{exp(y * log(x))}.
 The value returned for
-\tcode{pow(0,0)}
+\tcode{pow(0, 0)}
 is implementation-defined.
 \end{itemdescr}
 
 \indexlibrary{\idxcode{sin}!\idxcode{complex}}%
 \begin{itemdecl}
-template<class T> complex<T> sin  (const complex<T>& x);
+template<class T> complex<T> sin(const complex<T>& x);
 \end{itemdecl}
 
 \begin{itemdescr}
@@ -1196,7 +1190,7 @@
 
 \indexlibrary{\idxcode{sinh}!\idxcode{complex}}%
 \begin{itemdecl}
-template<class T> complex<T> sinh (const complex<T>& x);
+template<class T> complex<T> sinh(const complex<T>& x);
 \end{itemdecl}
 
 \begin{itemdescr}
@@ -1207,13 +1201,13 @@
 
 \indexlibrary{\idxcode{sqrt}!\idxcode{complex}}%
 \begin{itemdecl}
-template<class T> complex<T> sqrt (const complex<T>& x);
+template<class T> complex<T> sqrt(const complex<T>& x);
 \end{itemdecl}
 
 \begin{itemdescr}
 \pnum
 \notes
-the branch cuts are along the negative real axis.
+The branch cuts are along the negative real axis.
 
 \pnum
 \returns
@@ -1225,7 +1219,7 @@
 
 \indexlibrary{\idxcode{tan}!\idxcode{complex}}%
 \begin{itemdecl}
-template<class T> complex<T> tan  (const complex<T>& x);
+template<class T> complex<T> tan(const complex<T>& x);
 \end{itemdecl}
 
 \begin{itemdescr}
@@ -1236,7 +1230,7 @@
 
 \indexlibrary{\idxcode{tanh}!\idxcode{complex}}%
 \begin{itemdecl}
-template<class T> complex<T> tanh (const complex<T>& x);
+template<class T> complex<T> tanh(const complex<T>& x);
 \end{itemdecl}
 
 \begin{itemdescr}