Mathematics
f(z)    Complex Numbers and Functions     Tech Note
Complex Hyperbolic Functions

This page shows how to compute several complex hyperbolic functions, including hyperbolic cosine, hyperbolic sine, hyperbolic tangent, hyperbolic secant, hyperbolic cosecant, and hyperbolic cotangent.

CCosh:  Complex hyperbolic cosine

CCosh function

// z   = cosh(a)
FUNCTION CCosh (CONST a:  TComplex):  TComplex;
  VAR
    aTemp:  TComplex;
BEGIN  // Abramowitz formula 4.5.50 on p. 84
  aTemp := CConvert (a,cfRectangular);
  RESULT := CSet(COSH(aTemp.x)*COS(aTemp.y),
                 SINH(aTemp.x)*SIN(aTemp.y))
END {CCosh};

Examples

Complex hyperbolic cosine:  CCosh = Cosh(z)
 
                             COSH(z)                          COSH(z)
          z                rectangular                        polar
     ------------  ----------------------------   --------------------------------
 1    0.0 +  0.0i   1.000000000 +  0.000000000i   1.000000000 * CIS(  0.000000000)
 2    0.5 +  0.5i   0.989584883 +  0.249826398i   1.020632877 * CIS(  0.247288622)
 3   -0.5 +  0.5i   0.989584883 -  0.249826398i   1.020632877 * CIS( -0.247288622)
 4   -0.5 -  0.5i   0.989584883 +  0.249826398i   1.020632877 * CIS(  0.247288622)
 5    0.5 -  0.5i   0.989584883 -  0.249826398i   1.020632877 * CIS( -0.247288622)
 6    1.0 +  0.0i   1.543080635 +  0.000000000i   1.543080635 * CIS(  0.000000000)
 7    1.0 +  1.0i   0.833730025 +  0.988897706i   1.293454455 * CIS(  0.870327425)
 8    0.0 +  1.0i   0.540302306 +  0.000000000i   0.540302306 * CIS(  0.000000000)
 9   -1.0 +  1.0i   0.833730025 -  0.988897706i   1.293454455 * CIS( -0.870327425)
10   -1.0 +  0.0i   1.543080635 -  0.000000000i   1.543080635 * CIS(  0.000000000)
11   -1.0 -  1.0i   0.833730025 +  0.988897706i   1.293454455 * CIS(  0.870327425)
12    0.0 -  1.0i   0.540302306 +  0.000000000i   0.540302306 * CIS(  0.000000000)
13    1.0 -  1.0i   0.833730025 -  0.988897706i   1.293454455 * CIS( -0.870327425)
14    5.0 +  0.0i  74.209948525 +  0.000000000i  74.209948525 * CIS(  0.000000000)
15    5.0 +  3.0i -73.467292213 + 10.471557675i  74.209814346 * CIS(  3.000012685)
16    0.0 +  3.0i  -0.989992497 +  0.000000000i   0.989992497 * CIS(  3.141592654)
17   -5.0 +  3.0i -73.467292213 - 10.471557675i  74.209814346 * CIS( -3.000012685)
18   -5.0 +  0.0i  74.209948525 -  0.000000000i  74.209948525 * CIS(  0.000000000)
19   -5.0 -  3.0i -73.467292213 + 10.471557675i  74.209814346 * CIS(  3.000012685)
20    0.0 -  3.0i  -0.989992497 +  0.000000000i   0.989992497 * CIS( -3.141592654)
21   -5.0 -  3.0i -73.467292213 + 10.471557675i  74.209814346 * CIS(  3.000012685)

 

CSinh: Complex hyperbolic sine

CSinh function

// z   = sinh(a)
FUNCTION CSinh (CONST a:  TComplex):  TComplex;
  VAR
    aTemp:  TComplex;
BEGIN  // Abramowitz formula 4.5.49 on p.84
  aTemp := CConvert (a,cfRectangular);
  RESULT := CSet(SINH(aTemp.x)*COS(aTemp.y),
                 COSH(aTemp.x)*SIN(aTemp.y))
END {CSinh};

The table below, in addition to showing the complex sinh function, also shows the results of the computation, cosh2(z) - sinh2(z), which is from the mathematical identity cosh2(z) - sinh2(z) = 1.

Examples

Complex hyperbolic sine:  CSinh = Sinh(z)
 
                             SINH(z)               COSH^2(z) + SINH^2(z) = 1
          z                rectangular                    rectangular
     ------------  ----------------------------   ---------------------------
 1    0.0 +  0.0i   0.000000000 +  0.000000000i   1.000000000 +  0.000000000i
 2    0.5 +  0.5i   0.457304153 +  0.540612686i   1.000000000 +  0.000000000i
 3   -0.5 +  0.5i  -0.457304153 +  0.540612686i   1.000000000 +  0.000000000i
 4   -0.5 -  0.5i  -0.457304153 -  0.540612686i   1.000000000 +  0.000000000i
 5    0.5 -  0.5i   0.457304153 -  0.540612686i   1.000000000 +  0.000000000i
 6    1.0 +  0.0i   1.175201194 +  0.000000000i   1.000000000 +  0.000000000i
 7    1.0 +  1.0i   0.634963915 +  1.298457581i   1.000000000 +  0.000000000i
 8    0.0 +  1.0i   0.000000000 +  0.841470985i   1.000000000 +  0.000000000i
 9   -1.0 +  1.0i  -0.634963915 +  1.298457581i   1.000000000 +  0.000000000i
10   -1.0 +  0.0i  -1.175201194 +  0.000000000i   1.000000000 +  0.000000000i
11   -1.0 -  1.0i  -0.634963915 -  1.298457581i   1.000000000 +  0.000000000i
12    0.0 -  1.0i   0.000000000 -  0.841470985i   1.000000000 +  0.000000000i
13    1.0 -  1.0i   0.634963915 -  1.298457581i   1.000000000 +  0.000000000i
14    5.0 +  0.0i  74.203210578 +  0.000000000i   1.000000000 +  0.000000000i
15    5.0 +  3.0i -73.460621696 + 10.472508534i   1.000000000 +  0.000000000i
16    0.0 +  3.0i   0.000000000 +  0.141120008i   1.000000000 +  0.000000000i
17   -5.0 +  3.0i  73.460621696 + 10.472508534i   1.000000000 +  0.000000000i
18   -5.0 +  0.0i -74.203210578 +  0.000000000i   1.000000000 +  0.000000000i
19   -5.0 -  3.0i  73.460621696 - 10.472508534i   1.000000000 +  0.000000000i
20    0.0 -  3.0i   0.000000000 -  0.141120008i   1.000000000 +  0.000000000i
21   -5.0 -  3.0i  73.460621696 - 10.472508534i   1.000000000 +  0.000000000i

 

CTanh:  Complex hyperbolic tangent

CTanh function

// z   = tanh(a)
FUNCTION CTanh (CONST a:  TComplex):  TComplex;
  VAR
    aTemp:  TComplex;
    temp :  TReal;
BEGIN  // Abramowitz formula 4.5.51 on p. 84
  aTemp := CConvert (a,cfRectangular);
  temp := COSH(2.0*aTemp.x) + COS(2.0*aTemp.y);
  TRY
    RESULT := CSet(SINH(2.0*aTemp.x)/temp,
                   SIN(2.0*aTemp.y) /temp)
  EXCEPT
    RESULT := CSet( SignedInfinity(SINH(2.0*aTemp.x)),
                    SignedInfinity(SIN(2.0*aTemp.y)) );
  END
END {CTanh};

The table below show the computation of tanh(z) directly and via calculation as sinh(z)/cosh(z).

Examples

Complex hyperbolic tangent:  CTanh = Tanh(z) = Sinh(z)/Cosh(z)
 
                             TANH(z)                   SINH(z) / COSH(z)
          z                rectangular                   rectangular
     ------------  ----------------------------   ---------------------------
 1    0.0 +  0.0i   0.000000000 +  0.000000000i   0.000000000 +  0.000000000i
 2    0.5 +  0.5i   0.564083141 +  0.403896455i   0.564083141 +  0.403896455i
 3   -0.5 +  0.5i  -0.564083141 +  0.403896455i  -0.564083141 +  0.403896455i
 4   -0.5 -  0.5i  -0.564083141 -  0.403896455i  -0.564083141 -  0.403896455i
 5    0.5 -  0.5i   0.564083141 -  0.403896455i   0.564083141 -  0.403896455i
 6    1.0 +  0.0i   0.761594156 +  0.000000000i   0.761594156 +  0.000000000i
 7    1.0 +  1.0i   1.083923327 +  0.271752585i   1.083923327 +  0.271752585i
 8    0.0 +  1.0i   0.000000000 +  1.557407725i   0.000000000 +  1.557407725i
 9   -1.0 +  1.0i  -1.083923327 +  0.271752585i  -1.083923327 +  0.271752585i
10   -1.0 +  0.0i  -0.761594156 +  0.000000000i  -0.761594156 +  0.000000000i
11   -1.0 -  1.0i  -1.083923327 -  0.271752585i  -1.083923327 -  0.271752585i
12    0.0 -  1.0i   0.000000000 -  1.557407725i   0.000000000 -  1.557407725i
13    1.0 -  1.0i   1.083923327 -  0.271752585i   1.083923327 -  0.271752585i
14    5.0 +  0.0i   0.999909204 +  0.000000000i   0.999909204 +  0.000000000i
15    5.0 +  3.0i   0.999912820 -  0.000025369i   0.999912820 -  0.000025369i
16    0.0 +  3.0i   0.000000000 -  0.142546543i   0.000000000 -  0.142546543i
17   -5.0 +  3.0i  -0.999912820 -  0.000025369i  -0.999912820 -  0.000025369i
18   -5.0 +  0.0i  -0.999909204 +  0.000000000i  -0.999909204 +  0.000000000i
19   -5.0 -  3.0i  -0.999912820 +  0.000025369i  -0.999912820 +  0.000025369i
20    0.0 -  3.0i   0.000000000 +  0.142546543i   0.000000000 +  0.142546543i
21   -5.0 -  3.0i  -0.999912820 +  0.000025369i  -0.999912820 +  0.000025369i

 

CSech: Complex hyperbolic secant

CSech function

// z   = sech(a)
FUNCTION CSech (CONST a:  TComplex):  TComplex;
  VAR
    aTemp:  TComplex;
BEGIN  // Abramowitz formula 4.5.5 on p. 83
  aTemp := CCosh(a);
  TRY
    RESULT := CDiv(ComplexOne, aTemp)
  EXCEPT
    ON EComplexZeroDivide DO
       RESULT := CSet(PositiveInfinity, PositiveInfinity);
    ON EComplexInvalidOp DO
       RESULT := CSet(PositiveInfinity, PositiveInfinity);
  END
END {CSech};

The table below, in addition to showing the complex secant function, also shows the results of the mathematical identity tanh2(z) + sech2(z) = 1. [This identity is just another version of the more-recognized identity, cosh2(z) - sinh2(z) = 1. Dividing all the terms in this identity by cosh2(z), and rearranging the terms, results in the identity used below.]

Examples

Complex hyperbolic secant:  CSech = Sech(z)
 
                             SECH(z)               TANH^2(z) + SECH^2(z) = 1
          z                rectangular                    rectangular
     ------------  ----------------------------   ---------------------------
 1    0.0 +  0.0i   1.000000000 +  0.000000000i   1.000000000 +  0.000000000i
 2    0.5 +  0.5i   0.949978868 -  0.239827631i   1.000000000 +  0.000000000i
 3   -0.5 +  0.5i   0.949978868 +  0.239827631i   1.000000000 +  0.000000000i
 4   -0.5 -  0.5i   0.949978868 -  0.239827631i   1.000000000 +  0.000000000i
 5    0.5 -  0.5i   0.949978868 +  0.239827631i   1.000000000 +  0.000000000i
 6    1.0 +  0.0i   0.648054274 +  0.000000000i   1.000000000 +  0.000000000i
 7    1.0 +  1.0i   0.498337031 -  0.591083842i   1.000000000 +  0.000000000i
 8    0.0 +  1.0i   1.850815718 +  0.000000000i   1.000000000 +  0.000000000i
 9   -1.0 +  1.0i   0.498337031 +  0.591083842i   1.000000000 +  0.000000000i
10   -1.0 +  0.0i   0.648054274 +  0.000000000i   1.000000000 +  0.000000000i
11   -1.0 -  1.0i   0.498337031 -  0.591083842i   1.000000000 +  0.000000000i
12    0.0 -  1.0i   1.850815718 +  0.000000000i   1.000000000 +  0.000000000i
13    1.0 -  1.0i   0.498337031 +  0.591083842i   1.000000000 +  0.000000000i
14    5.0 +  0.0i   0.013475282 +  0.000000000i   1.000000000 +  0.000000000i
15    5.0 +  3.0i  -0.013340477 -  0.001901466i   1.000000000 +  0.000000000i
16    0.0 +  3.0i  -1.010108666 +  0.000000000i   1.000000000 +  0.000000000i
17   -5.0 +  3.0i  -0.013340477 +  0.001901466i   1.000000000 +  0.000000000i
18   -5.0 +  0.0i   0.013475282 +  0.000000000i   1.000000000 +  0.000000000i
19   -5.0 -  3.0i  -0.013340477 -  0.001901466i   1.000000000 +  0.000000000i
20    0.0 -  3.0i  -1.010108666 +  0.000000000i   1.000000000 +  0.000000000i
21   -5.0 -  3.0i  -0.013340477 -  0.001901466i   1.000000000 +  0.000000000i

 

CCsch: Complex hyperbolic cosecant

CCsch function

// z   = csch(a)
FUNCTION CCsch (CONST a:  TComplex):  TComplex;
  VAR
    temp:  TComplex;
BEGIN  // Abramowitz formula 4.5.4 on p. 83
  temp := CSinh(a);
  TRY
    RESULT := CDiv(ComplexOne, temp)
  EXCEPT
    ON EComplexZeroDivide DO
       RESULT := CSet(PositiveInfinity, PositiveInfinity);
    ON EComplexInvalidOp DO
       RESULT := CSet(PositiveInfinity, PositiveInfinity);
  END
END {CCsch};

The table below, in addition to showing the complex cosecant function, also shows the results of the mathematical identity coth2(z) - csch2(z) = 1. [This identity is just another version of the more-recognized identity, cosh2(z) - sinh2(z) = 1. Dividing all the terms in this identity by sinh2(z), and rearranging the terms, results in the identity used below.]

Examples

Complex hyperbolic cosecant:  CCsch = Csch(z)
 
                             CSCH(z)               COTH^2(z) - CSCH^2(z) = 1
          z                rectangular                     rectangular
     ------------  ----------------------------   ---------------------------
 1    0.0 +  0.0i           INF +          INFi
 2    0.5 +  0.5i   0.912074264 -  1.078229695i   1.000000000 +  0.000000000i
 3   -0.5 +  0.5i  -0.912074264 -  1.078229695i   1.000000000 +  0.000000000i
 4   -0.5 -  0.5i  -0.912074264 +  1.078229695i   1.000000000 +  0.000000000i
 5    0.5 -  0.5i   0.912074264 +  1.078229695i   1.000000000 +  0.000000000i
 6    1.0 +  0.0i   0.850918128 +  0.000000000i   1.000000000 +  0.000000000i
 7    1.0 +  1.0i   0.303931002 -  0.621518017i   1.000000000 +  0.000000000i
 8    0.0 +  1.0i   0.000000000 -  1.188395106i   1.000000000 +  0.000000000i
 9   -1.0 +  1.0i  -0.303931002 -  0.621518017i   1.000000000 +  0.000000000i
10   -1.0 +  0.0i  -0.850918128 +  0.000000000i   1.000000000 +  0.000000000i
11   -1.0 -  1.0i  -0.303931002 +  0.621518017i   1.000000000 +  0.000000000i
12    0.0 -  1.0i   0.000000000 +  1.188395106i   1.000000000 +  0.000000000i
13    1.0 -  1.0i   0.303931002 +  0.621518017i   1.000000000 +  0.000000000i
14    5.0 +  0.0i   0.013476506 +  0.000000000i   1.000000000 +  0.000000000i
15    5.0 +  3.0i  -0.013341591 -  0.001901970i   1.000000000 +  0.000000000i
16    0.0 +  3.0i   0.000000000 -  7.086167396i   1.000000000 +  0.000000000i
17   -5.0 +  3.0i   0.013341591 -  0.001901970i   1.000000000 +  0.000000000i
18   -5.0 +  0.0i  -0.013476506 +  0.000000000i   1.000000000 +  0.000000000i
19   -5.0 -  3.0i   0.013341591 +  0.001901970i   1.000000000 +  0.000000000i
20    0.0 -  3.0i   0.000000000 +  7.086167396i   1.000000000 +  0.000000000i
21   -5.0 -  3.0i   0.013341591 +  0.001901970i   1.000000000 +  0.000000000i

 

CCoth:  Complex hyperbolic cotangent

CCoth function

// z   = coth(a)
FUNCTION CCoth (CONST a:  TComplex):  TComplex;
  VAR
    aTemp:  TComplex;
    temp :  TReal;
BEGIN  // Abramowitz formula 4.5.52 on p. 84
  aTemp := CConvert (a, cfRectangular);
  temp := COSH(2.0*aTemp.x) - COS(2.0*aTemp.y);
  TRY
    RESULT := CSet(SINH(2.0*aTemp.x)/temp,
                  -SIN(2.0*aTemp.y)/temp)
  EXCEPT
    ON EZeroDivide DO
       RESULT := CSet( SignedInfinity(SINH(2.0*aTemp.x)),
                       SignedInfinity(-SIN(2.0*aTemp.y)) );
    ON EInvalidOp DO
       RESULT := CSet(PositiveInfinity, PositiveInfinity);
  END
END {CCoth};

 

The table below shows the computation of coth(z) directly and via calculation as cosh(z)/sinh(z).

Examples

Complex hyperbolic cotangent:  CCoth = Coth(z) = Cosh(z)/Sinh(z)
 
                             COTH(z)                  COSH(z) / SINH(z)
          z                rectangular                  rectangular
     ------------  ----------------------------   ---------------------------
 1    0.0 +  0.0i           INF +          INFi           INF +          INFi
 2    0.5 +  0.5i   1.171945145 -  0.839139579i   1.171945145 -  0.839139579i
 3   -0.5 +  0.5i  -1.171945145 -  0.839139579i  -1.171945145 -  0.839139579i
 4   -0.5 -  0.5i  -1.171945145 +  0.839139579i  -1.171945145 +  0.839139579i
 5    0.5 -  0.5i   1.171945145 +  0.839139579i   1.171945145 +  0.839139579i
 6    1.0 +  0.0i   1.313035285 +  0.000000000i   1.313035285 +  0.000000000i
 7    1.0 +  1.0i   0.868014143 -  0.217621562i   0.868014143 -  0.217621562i
 8    0.0 +  1.0i   0.000000000 -  0.642092616i   0.000000000 -  0.642092616i
 9   -1.0 +  1.0i  -0.868014143 -  0.217621562i  -0.868014143 -  0.217621562i
10   -1.0 +  0.0i  -1.313035285 +  0.000000000i  -1.313035285 +  0.000000000i
11   -1.0 -  1.0i  -0.868014143 +  0.217621562i  -0.868014143 +  0.217621562i
12    0.0 -  1.0i   0.000000000 +  0.642092616i   0.000000000 +  0.642092616i
13    1.0 -  1.0i   0.868014143 +  0.217621562i   0.868014143 +  0.217621562i
14    5.0 +  0.0i   1.000090804 +  0.000000000i   1.000090804 +  0.000000000i
15    5.0 +  3.0i   1.000087187 +  0.000025373i   1.000087187 +  0.000025373i
16    0.0 +  3.0i   0.000000000 +  7.015252551i   0.000000000 +  7.015252551i
17   -5.0 +  3.0i  -1.000087187 +  0.000025373i  -1.000087187 +  0.000025373i
18   -5.0 +  0.0i  -1.000090804 +  0.000000000i  -1.000090804 +  0.000000000i
19   -5.0 -  3.0i  -1.000087187 -  0.000025373i  -1.000087187 -  0.000025373i
20    0.0 -  3.0i   0.000000000 -  7.015252551i   0.000000000 -  7.015252551i
21   -5.0 -  3.0i  -1.000087187 -  0.000025373i  -1.000087187 -  0.000025373i

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Updated 18 Feb 2002


since 24 June 2001