# SLATEC Routines --- ZBESH ---

```*DECK ZBESH
SUBROUTINE ZBESH (ZR, ZI, FNU, KODE, M, N, CYR, CYI, NZ, IERR)
C***BEGIN PROLOGUE  ZBESH
C***PURPOSE  Compute a sequence of the Hankel functions H(m,a,z)
C            for superscript m=1 or 2, real nonnegative orders a=b,
C            b+1,... where b>0, and nonzero complex argument z.  A
C            scaling option is available to help avoid overflow.
C***LIBRARY   SLATEC
C***CATEGORY  C10A4
C***TYPE      COMPLEX (CBESH-C, ZBESH-C)
C***KEYWORDS  BESSEL FUNCTIONS OF COMPLEX ARGUMENT,
C             BESSEL FUNCTIONS OF THE THIRD KIND, H BESSEL FUNCTIONS,
C             HANKEL FUNCTIONS
C***AUTHOR  Amos, D. E., (SNL)
C***DESCRIPTION
C
C                      ***A DOUBLE PRECISION ROUTINE***
C         On KODE=1, ZBESH computes an N member sequence of complex
C         Hankel (Bessel) functions CY(L)=H(M,FNU+L-1,Z) for super-
C         script M=1 or 2, real nonnegative orders FNU+L-1, L=1,...,
C         N, and complex nonzero Z in the cut plane -piCBESH returns the scaled
C         functions
C
C            CY(L) = H(M,FNU+L-1,Z)*exp(-(3-2*M)*Z*i),  i**2=-1
C
C         which removes the exponential behavior in both the upper
C         and lower half planes.  Definitions and notation are found
C         in the NBS Handbook of Mathematical Functions (Ref. 1).
C
C         Input
C           ZR     - DOUBLE PRECISION real part of nonzero argument Z
C           ZI     - DOUBLE PRECISION imag part of nonzero argument Z
C           FNU    - DOUBLE PRECISION initial order, FNU>=0
C           KODE   - A parameter to indicate the scaling option
C                    KODE=1  returns
C                            CY(L)=H(M,FNU+L-1,Z), L=1,...,N
C                        =2  returns
C                            CY(L)=H(M,FNU+L-1,Z)*exp(-(3-2M)*Z*i),
C                            L=1,...,N
C           M      - Superscript of Hankel function, M=1 or 2
C           N      - Number of terms in the sequence, N>=1
C
C         Output
C           CYR    - DOUBLE PRECISION real part of result vector
C           CYI    - DOUBLE PRECISION imag part of result vector
C           NZ     - Number of underflows set to zero
C                    NZ=0    Normal return
C                    NZ>0    CY(L)=0 for NZ values of L (if M=1 and
C                            Im(Z)>0 or if M=2 and Im(Z)<0, then
C                            CY(L)=0 for L=1,...,NZ; in the com-
C                            plementary half planes, the underflows
C                            may not be in an uninterrupted sequence)
C           IERR   - Error flag
C                    IERR=0  Normal return     - COMPUTATION COMPLETED
C                    IERR=1  Input error       - NO COMPUTATION
C                    IERR=2  Overflow          - NO COMPUTATION
C                            (abs(Z) too small and/or FNU+N-1
C                            too large)
C                    IERR=3  Precision warning - COMPUTATION COMPLETED
C                            (Result has half precision or less
C                            because abs(Z) or FNU+N-1 is large)
C                    IERR=4  Precision error   - NO COMPUTATION
C                            (Result has no precision because
C                            abs(Z) or FNU+N-1 is too large)
C                    IERR=5  Algorithmic error - NO COMPUTATION
C                            (Termination condition not met)
C
C *Long Description:
C
C         The computation is carried out by the formula
C
C            H(m,a,z) = (1/t)*exp(-a*t)*K(a,z*exp(-t))
C                   t = (3-2*m)*i*pi/2
C
C         where the K Bessel function is computed as described in the
C         prologue to CBESK.
C
C         Exponential decay of H(m,a,z) occurs in the upper half z
C         plane for m=1 and the lower half z plane for m=2.  Exponential
C         growth occurs in the complementary half planes.  Scaling
C         by exp(-(3-2*m)*z*i) removes the exponential behavior in the
C         whole z plane as z goes to infinity.
C
C         For negative orders, the formula
C
C            H(m,-a,z) = H(m,a,z)*exp((3-2*m)*a*pi*i)
C
C         can be used.
C
C         In most complex variable computation, one must evaluate ele-
C         mentary functions.  When the magnitude of Z or FNU+N-1 is
C         large, losses of significance by argument reduction occur.
C         Consequently, if either one exceeds U1=SQRT(0.5/UR), then
C         losses exceeding half precision are likely and an error flag
C         IERR=3 is triggered where UR=MAX(D1MACH(4),1.0D-18) is double
C         precision unit roundoff limited to 18 digits precision.  Also,
C         if either is larger than U2=0.5/UR, then all significance is
C         lost and IERR=4.  In order to use the INT function, arguments
C         must be further restricted not to exceed the largest machine
C         integer, U3=I1MACH(9).  Thus, the magnitude of Z and FNU+N-1
C         is restricted by MIN(U2,U3).  In IEEE arithmetic, U1,U2, and
C         U3 approximate 2.0E+3, 4.2E+6, 2.1E+9 in single precision
C         and 4.7E+7, 2.3E+15 and 2.1E+9 in double precision.  This
C         makes U2 limiting in single precision and U3 limiting in
C         double precision.  This means that one can expect to retain,
C         in the worst cases on IEEE machines, no digits in single pre-
C         cision and only 6 digits in double precision.  Similar con-
C         siderations hold for other machines.
C
C         The approximate relative error in the magnitude of a complex
C         Bessel function can be expressed as P*10**S where P=MAX(UNIT
C         ROUNDOFF,1.0E-18) is the nominal precision and 10**S repre-
C         sents the increase in error due to argument reduction in the
C         elementary functions.  Here, S=MAX(1,ABS(LOG10(ABS(Z))),
C         ABS(LOG10(FNU))) approximately (i.e., S=MAX(1,ABS(EXPONENT OF
C         ABS(Z),ABS(EXPONENT OF FNU)) ).  However, the phase angle may
C         have only absolute accuracy.  This is most likely to occur
C         when one component (in magnitude) is larger than the other by
C         several orders of magnitude.  If one component is 10**K larger
C         than the other, then one can expect only MAX(ABS(LOG10(P))-K,
C         0) significant digits; or, stated another way, when K exceeds
C         the exponent of P, no significant digits remain in the smaller
C         component.  However, the phase angle retains absolute accuracy
C         because, in complex arithmetic with precision P, the smaller
C         component will not (as a rule) decrease below P times the
C         magnitude of the larger component.  In these extreme cases,
C         the principal phase angle is on the order of +P, -P, PI/2-P,
C         or -PI/2+P.
C
C***REFERENCES  1. M. Abramowitz and I. A. Stegun, Handbook of Mathe-
C                 matical Functions, National Bureau of Standards
C                 Applied Mathematics Series 55, U. S. Department
C                 of Commerce, Tenth Printing (1972) or later.
C               2. D. E. Amos, Computation of Bessel Functions of
C                 Complex Argument, Report SAND83-0086, Sandia National
C                 Laboratories, Albuquerque, NM, May 1983.
C               3. D. E. Amos, Computation of Bessel Functions of
C                 Complex Argument and Large Order, Report SAND83-0643,
C                 Sandia National Laboratories, Albuquerque, NM, May
C                 1983.
C               4. D. E. Amos, A Subroutine Package for Bessel Functions
C                 of a Complex Argument and Nonnegative Order, Report
C                 SAND85-1018, Sandia National Laboratory, Albuquerque,
C                 NM, May 1985.
C               5. D. E. Amos, A portable package for Bessel functions
C                 of a complex argument and nonnegative order, ACM
C                 Transactions on Mathematical Software, 12 (September
C                 1986), pp. 265-273.
C
C***ROUTINES CALLED  D1MACH, I1MACH, ZABS, ZACON, ZBKNU, ZBUNK, ZUOIK
C***REVISION HISTORY  (YYMMDD)
C   830501  DATE WRITTEN
C   890801  REVISION DATE from Version 3.2
C   910415  Prologue converted to Version 4.0 format.  (BAB)
C   920128  Category corrected.  (WRB)
C   920811  Prologue revised.  (DWL)
C***END PROLOGUE  ZBESH
```