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Clarify the return value range of log(x) #589

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Mar 3, 2016
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Clarify the return value range of log(x) #589

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176ac16
[complex.numbers] Make whitespace and capitalization more consistent
tkoeppe Dec 18, 2015
b1f17ea
[complex] Clarify the range of return values of log
tkoeppe Dec 19, 2015
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56 changes: 25 additions & 31 deletions source/numerics.tex
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Original file line number Diff line number Diff line change
Expand Up @@ -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:
Expand All @@ -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>&);
Expand Down Expand Up @@ -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}}%
Expand All @@ -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 $\pi$.
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Note that this is merely an editorial and typographic reformulation of the original wording. It is unclear how to reconcole this specification of log with Annex G of C11, which requires the phase the logarithm of a negative real number to depend on the sign of the imaginary part (which can be +/-0).

\end{itemdescr}

\indexlibrary{\idxcode{log10}!\idxcode{complex}}%
Expand All @@ -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}
Expand All @@ -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}
Expand All @@ -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
Expand All @@ -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}
Expand All @@ -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}
Expand Down