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Diffstat (limited to 'manual/math.texi')
| -rw-r--r-- | manual/math.texi | 328 |
1 files changed, 138 insertions, 190 deletions
diff --git a/manual/math.texi b/manual/math.texi index 78d567b367..e2adccddb3 100644 --- a/manual/math.texi +++ b/manual/math.texi @@ -146,10 +146,8 @@ You can also compute the value of pi with the expression @code{acos @comment math.h @comment ISO @deftypefun double sin (double @var{x}) -@end deftypefun -@deftypefun float sinf (float @var{x}) -@end deftypefun -@deftypefun {long double} sinl (long double @var{x}) +@deftypefunx float sinf (float @var{x}) +@deftypefunx {long double} sinl (long double @var{x}) These functions return the sine of @var{x}, where @var{x} is given in radians. The return value is in the range @code{-1} to @code{1}. @end deftypefun @@ -157,10 +155,8 @@ radians. The return value is in the range @code{-1} to @code{1}. @comment math.h @comment ISO @deftypefun double cos (double @var{x}) -@end deftypefun -@deftypefun float cosf (float @var{x}) -@end deftypefun -@deftypefun {long double} cosl (long double @var{x}) +@deftypefunx float cosf (float @var{x}) +@deftypefunx {long double} cosl (long double @var{x}) These functions return the cosine of @var{x}, where @var{x} is given in radians. The return value is in the range @code{-1} to @code{1}. @end deftypefun @@ -168,10 +164,8 @@ radians. The return value is in the range @code{-1} to @code{1}. @comment math.h @comment ISO @deftypefun double tan (double @var{x}) -@end deftypefun -@deftypefun float tanf (float @var{x}) -@end deftypefun -@deftypefun {long double} tanl (long double @var{x}) +@deftypefunx float tanf (float @var{x}) +@deftypefunx {long double} tanl (long double @var{x}) These functions return the tangent of @var{x}, where @var{x} is given in radians. @@ -189,16 +183,14 @@ either positive or negative @code{HUGE_VAL}. In many applications where @code{sin} and @code{cos} are used, the value for the same argument of both of these functions is used at the same time. Since the algorithm to compute these values is very similar for -both functions there is an additional function with computes both values +both functions there is an additional function which computes both values at the same time. @comment math.h @comment GNU @deftypefun void sincos (double @var{x}, double *@var{sinx}, double *@var{cosx}) -@end deftypefun -@deftypefun void sincosf (float @var{x}, float *@var{sinx}, float *@var{cosx}) -@end deftypefun -@deftypefun void sincosl (long double @var{x}, long double *@var{sinx}, long double *@var{cosx}) +@deftypefunx void sincosf (float @var{x}, float *@var{sinx}, float *@var{cosx}) +@deftypefunx void sincosl (long double @var{x}, long double *@var{sinx}, long double *@var{cosx}) These functions return the sine of @var{x} in @code{*@var{sinx}} and the cosine of @var{x} in @code{*@var{cos}}, where @var{x} is given in radians. Both values, @code{*@var{sinx}} and @code{*@var{cosx}}, are in @@ -207,53 +199,62 @@ the range of @code{-1} to @code{1}. @cindex complex trigonometric functions -The trigonometric functions are in mathematics not only on real numbers. -They can be extended to complex numbers and the @w{ISO C 9X} standard -introduces these variants in the standard math library. +The trigonometric functions are in mathematics not only defined on real +numbers. They can be extended to complex numbers and the @w{ISO C 9X} +standard introduces these variants in the standard math library. @comment complex.h @comment ISO @deftypefun {complex double} csin (complex double @var{z}) -@end deftypefun -@deftypefun {complex float} csinf (complex float @var{z}) -@end deftypefun -@deftypefun {complex long double} csinl (complex long double @var{z}) +@deftypefunx {complex float} csinf (complex float @var{z}) +@deftypefunx {complex long double} csinl (complex long double @var{z}) These functions return the complex sine of the complex value in @var{z}. The mathematical definition of the complex sine is -@smallexample -sin (z) = 1/(2*i) * (exp (z*i) - exp (-z*i)) -@end smallexample +@ifinfo +@math{sin (z) = 1/(2*i) * (exp (z*i) - exp (-z*i))}. +@end ifinfo +@iftex +@tex +$$\sin(z) = {1\over 2i} (e^{zi} - e^{-zi})$$ +@end tex +@end iftex @end deftypefun @comment complex.h @comment ISO @deftypefun {complex double} ccos (complex double @var{z}) -@end deftypefun -@deftypefun {complex float} ccosf (complex float @var{z}) -@end deftypefun -@deftypefun {complex long double} ccosl (complex long double @var{z}) +@deftypefunx {complex float} ccosf (complex float @var{z}) +@deftypefunx {complex long double} ccosl (complex long double @var{z}) These functions return the complex cosine of the complex value in @var{z}. The mathematical definition of the complex cosine is -@smallexample -cos (z) = 1/2 * (exp (z*i) + exp (-z*i)) -@end smallexample +@ifinfo +@math{cos (z) = 1/2 * (exp (z*i) + exp (-z*i))} +@end ifinfo +@iftex +@tex +$$\cos(z) = {1\over 2} (e^{zi} + e^{-zi})$$ +@end tex +@end iftex @end deftypefun @comment complex.h @comment ISO @deftypefun {complex double} ctan (complex double @var{z}) -@end deftypefun -@deftypefun {complex float} ctanf (complex float @var{z}) -@end deftypefun -@deftypefun {complex long double} ctanl (complex long double @var{z}) +@deftypefunx {complex float} ctanf (complex float @var{z}) +@deftypefunx {complex long double} ctanl (complex long double @var{z}) These functions return the complex tangent of the complex value in @var{z}. The mathematical definition of the complex tangent is -@smallexample -tan (z) = 1/i * (exp (z*i) - exp (-z*i)) / (exp (z*i) + exp (-z*i)) -@end smallexample +@ifinfo +@math{tan (z) = 1/i * (exp (z*i) - exp (-z*i)) / (exp (z*i) + exp (-z*i))} +@end ifinfo +@iftex +@tex +$$\tan(z) = {1\over i} {e^{zi} - e^{-zi}\over e^{zi} + e^{-zi}}$$ +@end tex +@end iftex @end deftypefun @@ -268,10 +269,8 @@ respectively. @comment math.h @comment ISO @deftypefun double asin (double @var{x}) -@end deftypefun -@deftypefun float asinf (float @var{x}) -@end deftypefun -@deftypefun {long double} asinl (long double @var{x}) +@deftypefunx float asinf (float @var{x}) +@deftypefunx {long double} asinl (long double @var{x}) These functions compute the arc sine of @var{x}---that is, the value whose sine is @var{x}. The value is in units of radians. Mathematically, there are infinitely many such values; the one actually returned is the @@ -285,10 +284,8 @@ over the domain @code{-1} to @code{1}. @comment math.h @comment ISO @deftypefun double acos (double @var{x}) -@end deftypefun -@deftypefun float acosf (float @var{x}) -@end deftypefun -@deftypefun {long double} acosl (long double @var{x}) +@deftypefunx float acosf (float @var{x}) +@deftypefunx {long double} acosl (long double @var{x}) These functions compute the arc cosine of @var{x}---that is, the value whose cosine is @var{x}. The value is in units of radians. Mathematically, there are infinitely many such values; the one actually @@ -303,10 +300,8 @@ over the domain @code{-1} to @code{1}. @comment math.h @comment ISO @deftypefun double atan (double @var{x}) -@end deftypefun -@deftypefun float atanf (float @var{x}) -@end deftypefun -@deftypefun {long double} atanl (long double @var{x}) +@deftypefunx float atanf (float @var{x}) +@deftypefunx {long double} atanl (long double @var{x}) These functions compute the arc tangent of @var{x}---that is, the value whose tangent is @var{x}. The value is in units of radians. Mathematically, there are infinitely many such values; the one actually @@ -317,10 +312,8 @@ returned is the one between @code{-pi/2} and @code{pi/2} @comment math.h @comment ISO @deftypefun double atan2 (double @var{y}, double @var{x}) -@end deftypefun -@deftypefun float atan2f (float @var{y}, float @var{x}) -@end deftypefun -@deftypefun {long double} atan2l (long double @var{y}, long double @var{x}) +@deftypefunx float atan2f (float @var{y}, float @var{x}) +@deftypefunx {long double} atan2l (long double @var{y}, long double @var{x}) This is the two argument arc tangent function. It is similar to computing the arc tangent of @var{y}/@var{x}, except that the signs of both arguments are used to determine the quadrant of the result, and @var{x} is @@ -347,10 +340,8 @@ which are usable with complex numbers. @comment complex.h @comment ISO @deftypefun {complex double} casin (complex double @var{z}) -@end deftypefun -@deftypefun {complex float} casinf (complex float @var{z}) -@end deftypefun -@deftypefun {complex long double} casinl (complex long double @var{z}) +@deftypefunx {complex float} casinf (complex float @var{z}) +@deftypefunx {complex long double} casinl (complex long double @var{z}) These functions compute the complex arc sine of @var{z}---that is, the value whose sine is @var{z}. The value is in units of radians. @@ -361,10 +352,8 @@ limitation on the argument @var{z}. @comment complex.h @comment ISO @deftypefun {complex double} cacos (complex double @var{z}) -@end deftypefun -@deftypefun {complex float} cacosf (complex float @var{z}) -@end deftypefun -@deftypefun {complex long double} cacosl (complex long double @var{z}) +@deftypefunx {complex float} cacosf (complex float @var{z}) +@deftypefunx {complex long double} cacosl (complex long double @var{z}) These functions compute the complex arc cosine of @var{z}---that is, the value whose cosine is @var{z}. The value is in units of radians. @@ -376,10 +365,8 @@ limitation on the argument @var{z}. @comment complex.h @comment ISO @deftypefun {complex double} catan (complex double @var{z}) -@end deftypefun -@deftypefun {complex float} catanf (complex float @var{z}) -@end deftypefun -@deftypefun {complex long double} catanl (complex long double @var{z}) +@deftypefunx {complex float} catanf (complex float @var{z}) +@deftypefunx {complex long double} catanl (complex long double @var{z}) These functions compute the complex arc tangent of @var{z}---that is, the value whose tangent is @var{z}. The value is in units of radians. @end deftypefun @@ -394,10 +381,8 @@ the value whose tangent is @var{z}. The value is in units of radians. @comment math.h @comment ISO @deftypefun double exp (double @var{x}) -@end deftypefun -@deftypefun float expf (float @var{x}) -@end deftypefun -@deftypefun {long double} expl (long double @var{x}) +@deftypefunx float expf (float @var{x}) +@deftypefunx {long double} expl (long double @var{x}) These functions return the value of @code{e} (the base of natural logarithms) raised to power @var{x}. @@ -408,10 +393,8 @@ magnitude of the result is too large to be representable. @comment math.h @comment ISO @deftypefun double exp10 (double @var{x}) -@end deftypefun -@deftypefun float exp10f (float @var{x}) -@end deftypefun -@deftypefun {long double} exp10l (long double @var{x}) +@deftypefunx float exp10f (float @var{x}) +@deftypefunx {long double} exp10l (long double @var{x}) These functions return the value of @code{10} raised to the power @var{x}. Mathematically, @code{exp10 (x)} is the same as @code{exp (x * log (10))}. @@ -422,10 +405,8 @@ magnitude of the result is too large to be representable. @comment math.h @comment ISO @deftypefun double exp2 (double @var{x}) -@end deftypefun -@deftypefun float exp2f (float @var{x}) -@end deftypefun -@deftypefun {long double} exp2l (long double @var{x}) +@deftypefunx float exp2f (float @var{x}) +@deftypefunx {long double} exp2l (long double @var{x}) These functions return the value of @code{2} raised to the power @var{x}. Mathematically, @code{exp2 (x)} is the same as @code{exp (x * log (2))}. @@ -437,10 +418,8 @@ magnitude of the result is too large to be representable. @comment math.h @comment ISO @deftypefun double log (double @var{x}) -@end deftypefun -@deftypefun float logf (floatdouble @var{x}) -@end deftypefun -@deftypefun {long double} logl (long double @var{x}) +@deftypefunx float logf (floatdouble @var{x}) +@deftypefunx {long double} logl (long double @var{x}) These functions return the natural logarithm of @var{x}. @code{exp (log (@var{x}))} equals @var{x}, exactly in mathematics and approximately in C. @@ -460,10 +439,8 @@ The argument is zero. The log of zero is not defined. @comment math.h @comment ISO @deftypefun double log10 (double @var{x}) -@end deftypefun -@deftypefun float log10f (float @var{x}) -@end deftypefun -@deftypefun {long double} log10l (long double @var{x}) +@deftypefunx float log10f (float @var{x}) +@deftypefunx {long double} log10l (long double @var{x}) These functions return the base-10 logarithm of @var{x}. Except for the different base, it is similar to the @code{log} function. In fact, @code{log10 (@var{x})} equals @code{log (@var{x}) / log (10)}. @@ -472,10 +449,8 @@ different base, it is similar to the @code{log} function. In fact, @comment math.h @comment ISO @deftypefun double log2 (double @var{x}) -@end deftypefun -@deftypefun float log2f (float @var{x}) -@end deftypefun -@deftypefun {long double} log2l (long double @var{x}) +@deftypefunx float log2f (float @var{x}) +@deftypefunx {long double} log2l (long double @var{x}) These functions return the base-2 logarithm of @var{x}. Except for the different base, it is similar to the @code{log} function. In fact, @code{log2 (@var{x})} equals @code{log (@var{x}) / log (2)}. @@ -484,10 +459,8 @@ different base, it is similar to the @code{log} function. In fact, @comment math.h @comment ISO @deftypefun double pow (double @var{base}, double @var{power}) -@end deftypefun -@deftypefun float powf (float @var{base}, float @var{power}) -@end deftypefun -@deftypefun {long double} powl (long double @var{base}, long double @var{power}) +@deftypefunx float powf (float @var{base}, float @var{power}) +@deftypefunx {long double} powl (long double @var{base}, long double @var{power}) These are general exponentiation functions, returning @var{base} raised to @var{power}. @@ -508,10 +481,8 @@ An underflow or overflow condition was detected in the result. @comment math.h @comment ISO @deftypefun double sqrt (double @var{x}) -@end deftypefun -@deftypefun float sqrtf (float @var{x}) -@end deftypefun -@deftypefun {long double} sqrtl (long double @var{x}) +@deftypefunx float sqrtf (float @var{x}) +@deftypefunx {long double} sqrtl (long double @var{x}) These functions return the nonnegative square root of @var{x}. The @code{sqrt} function fails, and sets @code{errno} to @code{EDOM}, if @@ -524,10 +495,8 @@ number. @comment math.h @comment BSD @deftypefun double cbrt (double @var{x}) -@end deftypefun -@deftypefun float cbrtf (float @var{x}) -@end deftypefun -@deftypefun {long double} cbrtl (long double @var{x}) +@deftypefunx float cbrtf (float @var{x}) +@deftypefunx {long double} cbrtl (long double @var{x}) These functions return the cube root of @var{x}. They cannot fail; every representable real value has a representable real cube root. @end deftypefun @@ -535,10 +504,8 @@ fail; every representable real value has a representable real cube root. @comment math.h @comment ISO @deftypefun double hypot (double @var{x}, double @var{y}) -@end deftypefun -@deftypefun float hypotf (float @var{x}, float @var{y}) -@end deftypefun -@deftypefun {long double} hypotl (long double @var{x}, long double @var{y}) +@deftypefunx float hypotf (float @var{x}, float @var{y}) +@deftypefunx {long double} hypotl (long double @var{x}, long double @var{y}) These functions return @code{sqrt (@var{x}*@var{x} + @var{y}*@var{y})}. (This is the length of the hypotenuse of a right triangle with sides of length @var{x} and @var{y}, or the distance @@ -550,10 +517,8 @@ much smaller. See also the function @code{cabs} in @ref{Absolute Value}. @comment math.h @comment ISO @deftypefun double expm1 (double @var{x}) -@end deftypefun -@deftypefun float expm1f (float @var{x}) -@end deftypefun -@deftypefun {long double} expm1l (long double @var{x}) +@deftypefunx float expm1f (float @var{x}) +@deftypefunx {long double} expm1l (long double @var{x}) These functions return a value equivalent to @code{exp (@var{x}) - 1}. It is computed in a way that is accurate even if the value of @var{x} is near zero---a case where @code{exp (@var{x}) - 1} would be inaccurate due @@ -563,10 +528,8 @@ to subtraction of two numbers that are nearly equal. @comment math.h @comment ISO @deftypefun double log1p (double @var{x}) -@end deftypefun -@deftypefun float log1pf (float @var{x}) -@end deftypefun -@deftypefun {long double} log1pl (long double @var{x}) +@deftypefunx float log1pf (float @var{x}) +@deftypefunx {long double} log1pl (long double @var{x}) This function returns a value equivalent to @w{@code{log (1 + @var{x})}}. It is computed in a way that is accurate even if the value of @var{x} is near zero. @@ -584,45 +547,51 @@ definition. @comment complex.h @comment ISO @deftypefun {complex double} cexp (complex double @var{z}) -@end deftypefun -@deftypefun {complex float} cexpf (complex float @var{z}) -@end deftypefun -@deftypefun {complex long double} cexpl (complex long double @var{z}) +@deftypefunx {complex float} cexpf (complex float @var{z}) +@deftypefunx {complex long double} cexpl (complex long double @var{z}) These functions return the value of @code{e} (the base of natural logarithms) raised to power of the complex value @var{z}. +@noindent Mathematically this corresponds to the value -@smallexample -exp (z) = exp (creal (z)) * (cos (cimag (z)) + I * sin (cimag (z))) -@end smallexample +@ifinfo +@math{exp (z) = exp (creal (z)) * (cos (cimag (z)) + I * sin (cimag (z)))} +@end ifinfo +@iftex +@tex +$$\exp(z) = e^z = e^{{\rm Re} z} (\cos ({\rm Im} z) + i \sin ({\rm Im} z))$$ +@end tex +@end iftex @end deftypefun @comment complex.h @comment ISO @deftypefun {complex double} clog (complex double @var{z}) -@end deftypefun -@deftypefun {complex float} clogf (complex float @var{z}) -@end deftypefun -@deftypefun {complex long double} clogl (complex long double @var{z}) +@deftypefunx {complex float} clogf (complex float @var{z}) +@deftypefunx {complex long double} clogl (complex long double @var{z}) These functions return the natural logarithm of the complex value @var{z}. Unlike the real value version @code{log} and its variants, @code{clog} has no limit for the range of its argument @var{z}. +@noindent Mathematically this corresponds to the value -@smallexample -log (z) = log (cabs (z)) + I * carg (z) -@end smallexample +@ifinfo +@math{log (z) = log (cabs (z)) + I * carg (z)} +@end ifinfo +@iftex +@tex +$$\log(z) = \log(|z|) + i \arg(z)$$ +@end tex +@end iftex @end deftypefun @comment complex.h @comment ISO @deftypefun {complex double} csqrt (complex double @var{z}) -@end deftypefun -@deftypefun {complex float} csqrtf (complex float @var{z}) -@end deftypefun -@deftypefun {complex long double} csqrtl (complex long double @var{z}) +@deftypefunx {complex float} csqrtf (complex float @var{z}) +@deftypefunx {complex long double} csqrtl (complex long double @var{z}) These functions return the complex root of the argument @var{z}. Unlike the @code{sqrt} function these functions do not have any restriction on the value of the argument. @@ -631,16 +600,19 @@ the value of the argument. @comment complex.h @comment ISO @deftypefun {complex double} cpow (complex double @var{base}, complex double @var{power}) -@end deftypefun -@deftypefun {complex float} cpowf (complex float @var{base}, complex float @var{power}) -@end deftypefun -@deftypefun {complex long double} cpowl (complex long double @var{base}, complex long double @var{power}) +@deftypefunx {complex float} cpowf (complex float @var{base}, complex float @var{power}) +@deftypefunx {complex long double} cpowl (complex long double @var{base}, complex long double @var{power}) These functions return the complex value @var{BASE} raised to the power of @var{power}. This is computed as -@smallexample -cpow (x, y) = cexp (y * clog (x)) -@end smallexample +@ifinfo +@math{cpow (x, y) = cexp (y * clog (x))} +@end ifinfo +@iftex +@tex +$${\rm cpow}(x, y) = e^{y \log(x)}$$ +@end tex +@end iftex @end deftypefun @@ -654,10 +626,8 @@ see @ref{Exponents and Logarithms}. @comment math.h @comment ISO @deftypefun double sinh (double @var{x}) -@end deftypefun -@deftypefun float sinhf (float @var{x}) -@end deftypefun -@deftypefun {long double} sinhl (long double @var{x}) +@deftypefunx float sinhf (float @var{x}) +@deftypefunx {long double} sinhl (long double @var{x}) These functions return the hyperbolic sine of @var{x}, defined mathematically as @w{@code{(exp (@var{x}) - exp (-@var{x})) / 2}}. The function fails, and sets @code{errno} to @code{ERANGE}, if the value of @@ -667,10 +637,8 @@ function fails, and sets @code{errno} to @code{ERANGE}, if the value of @comment math.h @comment ISO @deftypefun double cosh (double @var{x}) -@end deftypefun -@deftypefun float coshf (float @var{x}) -@end deftypefun -@deftypefun {long double} coshl (long double @var{x}) +@deftypefunx float coshf (float @var{x}) +@deftypefunx {long double} coshl (long double @var{x}) These function return the hyperbolic cosine of @var{x}, defined mathematically as @w{@code{(exp (@var{x}) + exp (-@var{x})) / 2}}. The function fails, and sets @code{errno} to @code{ERANGE}, if the value @@ -680,10 +648,8 @@ of @var{x} is too large; that is, if overflow occurs. @comment math.h @comment ISO @deftypefun double tanh (double @var{x}) -@end deftypefun -@deftypefun float tanhf (float @var{x}) -@end deftypefun -@deftypefun {long double} tanhl (long double @var{x}) +@deftypefunx float tanhf (float @var{x}) +@deftypefunx {long double} tanhl (long double @var{x}) These functions return the hyperbolic tangent of @var{x}, whose mathematical definition is @w{@code{sinh (@var{x}) / cosh (@var{x})}}. @end deftypefun @@ -698,10 +664,8 @@ library are optimized for accuracy and speed. @comment complex.h @comment ISO @deftypefun {complex double} csinh (complex double @var{z}) -@end deftypefun -@deftypefun {complex float} csinhf (complex float @var{z}) -@end deftypefun -@deftypefun {complex long double} csinhl (complex long double @var{z}) +@deftypefunx {complex float} csinhf (complex float @var{z}) +@deftypefunx {complex long double} csinhl (complex long double @var{z}) These functions return the complex hyperbolic sine of @var{z}, defined mathematically as @w{@code{(exp (@var{z}) - exp (-@var{z})) / 2}}. The function fails, and sets @code{errno} to @code{ERANGE}, if the value of @@ -711,10 +675,8 @@ result is too large. @comment complex.h @comment ISO @deftypefun {complex double} ccosh (complex double @var{z}) -@end deftypefun -@deftypefun {complex float} ccoshf (complex float @var{z}) -@end deftypefun -@deftypefun {complex long double} ccoshl (complex long double @var{z}) +@deftypefunx {complex float} ccoshf (complex float @var{z}) +@deftypefunx {complex long double} ccoshl (complex long double @var{z}) These functions return the complex hyperbolic cosine of @var{z}, defined mathematically as @w{@code{(exp (@var{z}) + exp (-@var{z})) / 2}}. The function fails, and sets @code{errno} to @code{ERANGE}, if the value of @@ -724,10 +686,8 @@ result is too large. @comment complex.h @comment ISO @deftypefun {complex double} ctanh (complex double @var{z}) -@end deftypefun -@deftypefun {complex float} ctanhf (complex float @var{z}) -@end deftypefun -@deftypefun {complex long double} ctanhl (complex long double @var{z}) +@deftypefunx {complex float} ctanhf (complex float @var{z}) +@deftypefunx {complex long double} ctanhl (complex long double @var{z}) These functions return the complex hyperbolic tangent of @var{z}, whose mathematical definition is @w{@code{csinh (@var{z}) / ccosh (@var{z})}}. @end deftypefun @@ -738,10 +698,8 @@ mathematical definition is @w{@code{csinh (@var{z}) / ccosh (@var{z})}}. @comment math.h @comment ISO @deftypefun double asinh (double @var{x}) -@end deftypefun -@deftypefun float asinhf (float @var{x}) -@end deftypefun -@deftypefun {long double} asinhl (long double @var{x}) +@deftypefunx float asinhf (float @var{x}) +@deftypefunx {long double} asinhl (long double @var{x}) These functions return the inverse hyperbolic sine of @var{x}---the value whose hyperbolic sine is @var{x}. @end deftypefun @@ -749,10 +707,8 @@ value whose hyperbolic sine is @var{x}. @comment math.h @comment ISO @deftypefun double acosh (double @var{x}) -@end deftypefun -@deftypefun float acoshf (float @var{x}) -@end deftypefun -@deftypefun {long double} acoshl (long double @var{x}) +@deftypefunx float acoshf (float @var{x}) +@deftypefunx {long double} acoshl (long double @var{x}) These functions return the inverse hyperbolic cosine of @var{x}---the value whose hyperbolic cosine is @var{x}. If @var{x} is less than @code{1}, @code{acosh} returns @code{HUGE_VAL}. @@ -761,10 +717,8 @@ value whose hyperbolic cosine is @var{x}. If @var{x} is less than @comment math.h @comment ISO @deftypefun double atanh (double @var{x}) -@end deftypefun -@deftypefun float atanhf (float @var{x}) -@end deftypefun -@deftypefun {long double} atanhl (long double @var{x}) +@deftypefunx float atanhf (float @var{x}) +@deftypefunx {long double} atanhl (long double @var{x}) These functions return the inverse hyperbolic tangent of @var{x}---the value whose hyperbolic tangent is @var{x}. If the absolute value of @var{x} is greater than or equal to @code{1}, @code{atanh} returns @@ -776,10 +730,8 @@ value whose hyperbolic tangent is @var{x}. If the absolute value of @comment complex.h @comment ISO @deftypefun {complex double} casinh (complex double @var{z}) -@end deftypefun -@deftypefun {complex float} casinhf (complex float @var{z}) -@end deftypefun -@deftypefun {complex long double} casinhl (complex long double @var{z}) +@deftypefunx {complex float} casinhf (complex float @var{z}) +@deftypefunx {complex long double} casinhl (complex long double @var{z}) These functions return the inverse complex hyperbolic sine of @var{z}---the value whose complex hyperbolic sine is @var{z}. @end deftypefun @@ -787,10 +739,8 @@ These functions return the inverse complex hyperbolic sine of @comment complex.h @comment ISO @deftypefun {complex double} cacosh (complex double @var{z}) -@end deftypefun -@deftypefun {complex float} cacoshf (complex float @var{z}) -@end deftypefun -@deftypefun {complex long double} cacoshl (complex long double @var{z}) +@deftypefunx {complex float} cacoshf (complex float @var{z}) +@deftypefunx {complex long double} cacoshl (complex long double @var{z}) These functions return the inverse complex hyperbolic cosine of @var{z}---the value whose complex hyperbolic cosine is @var{z}. Unlike the real valued function @code{acosh} there is not limit for the range @@ -800,10 +750,8 @@ of the argument. @comment complex.h @comment ISO @deftypefun {complex double} catanh (complex double @var{z}) -@end deftypefun -@deftypefun {complex float} catanhf (complex float @var{z}) -@end deftypefun -@deftypefun {complex long double} catanhl (complex long double @var{z}) +@deftypefunx {complex float} catanhf (complex float @var{z}) +@deftypefunx {complex long double} catanhl (complex long double @var{z}) These functions return the inverse complex hyperbolic tangent of @var{z}---the value whose complex hyperbolic tangent is @var{z}. Unlike the real valued function @code{atanh} there is not limit for the range |