## N1886: Complex math functions cacosh and ctanh

Submitter: Fred J. Tydeman
Submission Date: 2014-10-28

Summary

Complex math functions (cacosh (G.6.2.1) and ctanh(G.6.2.6)) are incorrectly specified.
1. cacosh( 0.0 + I*NaN ) should be NaN + I*pi/2 (not NaN + I*NaN).
2. Reasons: Mathematically, cacosh(0.0+I*y) = asinh(y) + I*pi/2. Also, C requires cacos(0+I*NaN) to be pi/2+I*NAN, which along with the mathematically identity cacosh(z) = +/-I * cacos(z), means cacosh(0.0 + I*NaN) is NaN + I*pi/2.

3. ctanh(+0.0+I*NaN) should be 0.0 + I*NaN (not NaN+I*NaN)
4. ctanh(+0.0+I*INF) should be 0.0 + I*NaN w/ invalid (not NaN+I*NaN w/ invalid)
5. Reason for above two: Since ctanh(x+I*y) = (sinh(2x) + I*sin(2y)) / (cosh(2x) + cos(2y)), for any rational number y, cos(2y) cannot be exactly -1, so no 0/(1+(-1)),so no 0/0, so no NaN for the real component of the result

Suggested Technical Corrigendum

Add to G.6.2.1 cacosh before 4th bullet: cacosh(0.0+I*NaN) returns NaN + I*pi/2

Add to G.6.2.1 cacosh 4th bullet: "non-zero" so it reads: cacosh(x + iNaN) returns NaN + i*NaN and optionally raises the ''invalid'' floating-point exception, for finite non-zero x.

Add to G.6.2.6 ctanh before 3rd bullet: ctanh(0.0+I*INF) returns 0.0+I*NAN and raises the ''invalid'' floating-point exception.

Add to G.6.2.6 ctanh 3rd bullet: "non-zero" so it reads: ctanh(x + I*INF) returns NaN + i*NaN and raises the ''invalid'' floating-point exception, for finite non-zero x.

Add to G.6.2.6 ctanh before 4th bullet: ctanh(0.0+I*NaN) returns 0.0+I*NAN

Add to G.6.2.6 ctanh 4th bullet: "non-zero" so it reads: ctanh(x + I*NAN) returns NaN + i*NaN and optionally raises the ''invalid'' floating-point exception, for finite non-zero x.