From 754bbf7a25a8dda49b5d08ef0d0443bbf5af0e36 Mon Sep 17 00:00:00 2001 From: Craig Jennings Date: Sun, 7 Apr 2024 13:41:34 -0500 Subject: new repository --- devdocs/c/numeric%2Fcomplex%2Fcatanh.html | 78 +++++++++++++++++++++++++++++++ 1 file changed, 78 insertions(+) create mode 100644 devdocs/c/numeric%2Fcomplex%2Fcatanh.html (limited to 'devdocs/c/numeric%2Fcomplex%2Fcatanh.html') diff --git a/devdocs/c/numeric%2Fcomplex%2Fcatanh.html b/devdocs/c/numeric%2Fcomplex%2Fcatanh.html new file mode 100644 index 00000000..23fd8cf7 --- /dev/null +++ b/devdocs/c/numeric%2Fcomplex%2Fcatanh.html @@ -0,0 +1,78 @@ +

catanhf, catanh, catanhl

Defined in header <complex.h> 
float complex       catanhf( float complex z );
+		 (1) (since C99) 
double complex      catanh( double complex z );
+		 (2) (since C99) 
long double complex catanhl( long double complex z );
+		 (3) (since C99)
Defined in header <tgmath.h> 
#define atanh( z )
+		 (4) (since C99)
+1-3) Computes the complex arc hyperbolic tangent of z with branch cuts outside the interval [−1; +1] along the real axis.
+4) Type-generic macro: If z has type long double complex, catanhl is called. if z has type double complex, catanh is called, if z has type float complex, catanhf is called. If z is real or integer, then the macro invokes the corresponding real function (atanhf, atanh, atanhl). If z is imaginary, then the macro invokes the corresponding real version of atan, implementing the formula atanh(iy) = i atan(y), and the return type is imaginary.

Parameters

+ +
z - complex argument

Return value

If no errors occur, the complex arc hyperbolic tangent of z is returned, in the range of a half-strip mathematically unbounded along the real axis and in the interval [−iπ/2; +iπ/2] along the imaginary axis.

+

Error handling and special values

Errors are reported consistent with math_errhandling

+

If the implementation supports IEEE floating-point arithmetic,

+

Notes

Although the C standard names this function "complex arc hyperbolic tangent", the inverse functions of the hyperbolic functions are the area functions. Their argument is the area of a hyperbolic sector, not an arc. The correct name is "complex inverse hyperbolic tangent", and, less common, "complex area hyperbolic tangent".

+

Inverse hyperbolic tangent is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segmentd (-∞,-1] and [+1,+∞) of the real axis. The mathematical definition of the principal value of the inverse hyperbolic tangent is atanh z =

+ln(1+z)-ln(z-1)/2.


For any z, atanh(z) =

+atan(iz)/i

Example

#include <stdio.h>
+#include <complex.h>
+ 
+int main(void)
+{
+    double complex z = catanh(2);
+    printf("catanh(+2+0i) = %f%+fi\n", creal(z), cimag(z));
+ 
+    double complex z2 = catanh(conj(2)); // or catanh(CMPLX(2, -0.0)) in C11
+    printf("catanh(+2-0i) (the other side of the cut) = %f%+fi\n", creal(z2), cimag(z2));
+ 
+    // for any z, atanh(z) = atan(iz)/i
+    double complex z3 = catanh(1+2*I);
+    printf("catanh(1+2i) = %f%+fi\n", creal(z3), cimag(z3));
+    double complex z4 = catan((1+2*I)*I)/I;
+    printf("catan(i * (1+2i))/i = %f%+fi\n", creal(z4), cimag(z4));
+}

Output:

+
catanh(+2+0i) = 0.549306+1.570796i
+catanh(+2-0i) (the other side of the cut) = 0.549306-1.570796i
+catanh(1+2i) = 0.173287+1.178097i
+catan(i * (1+2i))/i = 0.173287+1.178097i

References

See also

+ + + + +
+
(C99)(C99)(C99)
 computes the complex arc hyperbolic sine
(function)
+
(C99)(C99)(C99)
 computes the complex arc hyperbolic cosine
(function)
+
(C99)(C99)(C99)
 computes the complex hyperbolic tangent
(function)
+
(C99)(C99)(C99)
 computes inverse hyperbolic tangent (\({\small\operatorname{artanh}{x} }\)artanh(x))
(function)
C++ documentation for atanh
+

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Licensed under the Creative Commons Attribution-ShareAlike Unported License v3.0.
+ https://en.cppreference.com/w/c/numeric/complex/catanh +

+
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