TANH

This function extends FORTRAN’s generic function TANH to evaluate the complex hyperbolic tangent.

Required Arguments

Z — Complex number representing the angle in radians for which the hyperbolic tangent is desired. (Input)

FORTRAN 90 Interface

Generic: TANH (Z)

Specific: The specific interface names are CTANH and ZTANH.

FORTRAN 77 Interface

Complex: CTANH (Z)

Double complex: The double complex function name is ZTANH.

Let z = x + iy. If ∣cosh z∣2 is very small, that is, if y mod π is very close to π/2 or 3π/2 and if x is small, then tanh z is nearly singular; a fatal error condition is reported. If ∣cosh z∣2 is somewhat larger but still small, then the result will be less accurate than half precision. When 2y (z = x + iy) is so large that sin 2y cannot be evaluated accurately to even zero precision, the following situation results. If ∣x∣ < 3/2, then TANH cannot be evaluated accurately to better than one significant figure. If 3/2 ≤∣y∣ < –1/2 ln (ɛ/2), then TANH can be evaluated by ignoring the imaginary part of the argument; however, the answer will be less accurate than half precision. Here, ɛ = AMACH(4) is the machine precision.

Example

In this example, tanh(1 + i) is computed and printed.

USE TANH_INT

USE UMACH_INT

IMPLICIT NONE

! Declare variables

INTEGER NOUT

COMPLEX VALUE, Z

! Compute

Z = (1.0, 1.0)

VALUE = TANH(Z)

! Print the results

CALL UMACH (2, NOUT)

WRITE (NOUT,99999) Z, VALUE

99999 FORMAT (' TANH((', F6.3, ',', F6.3, ')) = (',&

F6.3, ',', F6.3, ')')

END

Output

TANH(( 1.000, 1.000)) = ( 1.084, 0.272)