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Some complex functions may be computed accurately from the basic
definition. But, preserving relative accuracy over the complete
domain of a few complex functions requires special techniques.
Note that the domain and range of complex functions may not be
obvious to the average user.
First, look at the mappings from domain z1 to range z2 for some
complex functions:
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Complex Function Screen Shots
Various identities for elementary functions in the complex plane,
including implementing the complex function using only real functions.
Let z = x + i y
then arg z = arctan y/x real function using signs of x and y
modulus z = sqrt(x*x+y*y) = |z| length of complex vector
re z = x
im z = y
SQRT
i (arg z)/2
sqrt z = sqrt(|z|) e
thus yielding half the angle with magnitude sqrt(|z|)
modulus z + re z modulus z - re z
sqrt z = sqrt ---------------- +/- i sqrt ----------------
2 2
where the sign of the imaginary part of the result is
the sign of the imaginary part of z
2 3
z z z
sqrt 1+z = 1 + - - -- + -- - ...
2 8 16
sqrt( 0 + i0) = 0 + i0
sqrt(-0 + i0) = 0 + i0
sqrt( 0 - i0) = 0 - i0
sqrt(-0 - i0) = 0 - i0
sqrt(z)**2 = z
sqrt(z*z) = z re z > 0
sqrt(1/z) = 1/sqrt(z) not for -0 or negative real axis
conjugate(sqrt(z)) = sqrt(conjugate(z)) not for -0 or negative real axis
Branch cut:
The branch cut lies along the negative real axis.
Domain:
Mathematically unbounded
Range:
Right half plane including the imaginary axis. The algebraic sign
of the real part is positive.
The sign of the imaginary part of the result is the same as the
sign of im z.
LOG
log z = ln( modulus z) + i argument z
2 3
z z
log 1+z = z - -- + -- - ... for |z| < 1
2 3
log(exp(z)) = z im z in [-Pi,Pi]
log(1/z) = -log(z) not on negative real axis
conjugate(log(z)) = log(conjugate(z)) not on negative real axis
Branch cut:
The branch cut lies along the negative real axis.
Domain:
Modulus z /= 0.0
Range:
Real part mathematically unbounded, imaginary part in range - Pi to Pi
The sign of the imaginary part of the result is the same as the
sign of im z.
EXP
z re z re z
e = e cos (im z) + i e sin (im z)
i x
e = cos(x) + i sin(x)
2 3
z z z
e = 1 + z + -- + -- + ...
2! 3!
z z+i 2Pi
e = e periodic with period i 2Pi
exp(-z) = 1/exp(z) not on negative real axis
exp(log(z)) = z |z| non zero
conjugate(exp(z)) = exp(conjugate(z))
Branch cut:
None
Domain:
re z < ln 'LARGE
Range:
Mathematically unbounded
For modulus z = 0.0, the result is 1.0 + z
"**"
w (w * ln z)
z = e
2 3
w (w ln z) (w ln z)
z = 1 + w ln z + --------- + --------- + ... for |z| non zero
2! 3!
w
log(z ) = w * log(z)
SIN
sin z = sin(re z) cosh(im z) + i cos(re z) sinh(im z)
iz -iz
e - e
sin z = -----------
2i
3 5
z z
sin z = z - -- + -- - ...
3! 5!
sin z = cos(z - Pi/2) periodic with period Pi
sin(z) = sin(z+2Pi)
sin z = -i sinh iz
sin(arcsin(z)) = z
sin(z) = -sin(-z)
conjugate(sin(z)) = sin(conjugate(z))
Domain:
|im z| < ln 'LARGE
COS
cos z = cos(re z) cosh(im z) - i sin(re z) sinh(im z)
iz -iz
e + e
cos z = -----------
2
2 4
z z
cos z = 1 - -- + -- - ...
2! 4!
cos z = sin(z + Pi/2) periodic with period Pi
cos(z) = cos(z+2Pi)
cos(z) = cos(-z)
cos z = cosh iz
cos(arccos(z)) = z
conjugate(cos(z)) = cos(conjugate(z))
Domain:
|im z| < ln 'LARGE
TAN
tan z = sin z / cos z
iz -iz
e - e
tan z = -i ---------- limit = -i when im z > ln 'LARGE
iz -iz limit = i when im z < - ln 'LARGE
e + e
3 5 7
z 2z 17z
tan z = z + -- + --- + ---- + ... for |z| < 1
3 15 315
sin x cos x sinh y cosh y
tan z = ---------------- +i ----------------
2 2 2 2
cos x + sinh y cos x + sinh y
tan z = cot(Pi/2 - z) periodic with period Pi
tan z = tan(z+2Pi)
tan z = 1/cot z
tan(z) = -tan(-z)
tan z = -i tanh iz
tan(arctan(z))=z
conjugate(tan(z)) = tan(conjugate(z))
Branch cut:
None
Domain:
Mathematically unbounded
Range:
Mathematically unbounded
For modulus z = 0.0, the result is z
COT
cot z = cos z / sin z
iz -iz
e + e
cot z = i ---------- limit = i when im z > ln 'LARGE
iz -iz limit = -i when im z < -ln 'LARGE
e - e
sin x cos x sinh y cosh y
cot z = ---------------- -i ----------------
2 2 2 2
sin x + sinh y sin x + sinh y
3 5
1 z z 2z
cot z = - - - - -- - --- - ...
z 3 45 945
cot z = tan(Pi/2 - z) periodic with period Pi
cot(z) = cot(z+2Pi)
cot(z) = -cot(-z)
cot z = 1/tan z
conjugate(cot(z)) = cot(conjugate(z))
Branch cut:
None
Domain:
Mathematically unbounded
Range:
Mathematically unbounded
For modulus z = 0.0, the result is z
ARCSIN
2
arcsin z = - i ln(i z + sqrt(1-z ))
arcsin z = - i ln(i z + sqrt(1-z)*sqrt(1+z))
3 5 7
z 3z 5z
arcsin z = z + -- + --- + ---- + ... for |z| < 1
6 40 112
1 3 15
arcsin z = -i( ln(2iz) - --- - ---- - ----- - ... ) for |z| > 1
2 4 6
4z 32z 288z
2
arcsin z = arctan(z/sqrt(1-z )) fix re of result
arcsin z =
2 2 1/2 2 2 1/2
arcsin(1/2 (x + 2 x + 1 + y ) - 1/2 (x - 2 x + 1 + y ) )
+ i
2 2 1/2 2 2 1/2
csgn(-i x + y) ln(1/2 (x + 2 x + 1 + y ) + 1/2 (x - 2 x + 1 + y )
2 2 1/2 2 2 1/2 2 1/2
+ ((1/2 (x + 2 x + 1 + y ) + 1/2 (x - 2 x + 1 + y ) ) - 1) )
note: The csgn function is used to determine in which half-plane
(`left' or `right') the complex-valued expression
or number x lies. It is defined by
/ 1 if Re(x) > 0 or Re(x) = 0 and Im(x) > 0
csgn(x) = < -1 if Re(x) < 0 or Re(x) = 0 and Im(x) < 0
\ 0 if x = 0
arcsin z = pi/2 - arccos z
arcsin z = -i arcsinh iz
arcsin(sin(z)) = z
Branch cut:
The real axis not in [ -1.0, 1.0 ]
Domain:
Mathematically unbounded
Range:
Imaginary part mathematically unbounded, real part in [ -Pi/2, Pi/2 ]
For modulus z = 0.0, the result is z
ARCCOS
1+z 1-z
arccos z = -i 2 ln( sqrt --- +i sqrt --- )
2 2
2
arccos z = -i ln(z + i sqrt(1-z ) )
3 5
Pi z 3z
arccos z = -- - z - -- - -- - ... for |z| < 1
2 6 40
arccos z = -i( ln(2z) for |z| > 1/sqrt(epsilon)
arccos z =
2 2 1/2 2 2 1/2
arccos(1/2 (x + 2 x + 1 + y ) - 1/2 (x - 2 x + 1 + y ) )
+ i
2 2 1/2 2 2 1/2
csgn(I x - y) ln(1/2 (x + 2 x + 1 + y ) + 1/2 (x - 2 x + 1 + y )
2 2 1/2 2 2 1/2 2 1/2
+ ((1/2 (x + 2 x + 1 + y ) + 1/2 (x - 2 x + 1 + y ) ) - 1) )
2
arccos z = arctan(sqrt(1-z )/z) fix re of result
arccos z = pi/2 - arcsin z
arccos(cos(z)) = z
Branch cut:
The real axis not in [ -1.0, 1.0 ]
Domain:
Mathematically unbounded
Range:
Imaginary part mathematically unbounded, real part in [ 0.0, Pi ]
ARCTAN
arctan z = -i ( ln(1 + i z) - ln(1 - i z) )/2
i i+z
arctan z = - ln --- must be fixed on slit for iz < -1
2 i-z
3 5 7
z z z
arctan z = z - -- + -- - -- + ... for |z| < 1
3 5 7
arctan z = Pi/2 - arccot z
arctan z = arccot(1/z)
arctan z = -i arctanh iz
arctan(tan(z)) = z
arctan z = 1/2 arctan(x, 1 - y) - 1/2 arctan(- x, y + 1)
2 2
x + (y + 1)
+i 1/4 ln(-------------)
2 2
x + (y - 1)
ARCCOT
i z-i
arccot z = - ln ---
2 z+i
3 5
Pi z z
arccot z = -- - z + -- - -- + ... for |z| < 1
2 3 5
arccot z = Pi/2 - arctan z
arccot z = arctan(1/z)
arccot(cot(z)) = z
arccot z = 1/2 Pi - 1/2 arctan(x, 1 - y) + 1/2 arctan(- x, y + 1)
2 2
x + (y + 1)
-i 1/4 ln(-------------)
2 2
x + (y - 1)
Range:
Imaginary part mathematically unbounded, real part in [0.0 , Pi]
SINH
sinh z = sinh(re z) cos(im z) + i cosh(re z) sin(im z)
z -z
e - e
sinh z = --------
2
3 5
z z
sinh z = z + -- + -- + ...
3! 5!
sinh z = -i cosh(z +i Pi/2) periodic with period i Pi
sinh z = -i sin iz
COSH
cosh z = cosh(re z) cos(im z) + i sinh(re z) sin(im z)
z -z
e + e
cosh z = --------
2
2 4
z z
cosh z = 1 + -- + -- + ...
2! 4!
cosh z = -i sinh(z +i Pi/2) periodic with period i Pi
cosh z = cos iz
TANH
tanh z = sinh z / cosh z
z -z
e - e
tanh z = --------
z -z
e + e
3 5 7
z 2z 17z
tanh z = z - -- + -- - ---- + ... for |z| < 1
3 15 315
tanh z = -i coth(z +i Pi/2) periodic with period i Pi
tanh z = 1/coth z
tanh z = -i tan iz
sinh x cosh x sin y cos y
tanh z = ---------------- +i ----------------
2 2 2 2
sinh x + cos y sinh x + cos y
COTH
coth z = cosh z / sinh z
z -z
e + e
coth z = --------
z -z
e - e
3 5
1 z z 2z
coth z = - + - - -- + --- - ... for |z| < 1
z 3 45 945
coth z = -i tanh(z +i Pi/2)
coth z = 1/tanh z
sinh x cosh x sin y cos y
coth z = ---------------- -i ----------------
2 2 2 2
sinh x + sin y sinh x + sin y
ARCSINH
2
arcsinh z = ln(z + sqrt(1 + z ) )
3 5 7
z 3z 5z
arcsinh z = z - -- + --- - --- + ... for |z| < 1
6 40 112
1 3 15
arcsinh z = ln(2z) + --- - ---- + ----- - ... for |z| > 1
2 4 6
4z 32z 288z
arcsinh z = -i arcsin iz
arcsinh(sinh(z)) = z
arcsinh z =
2 2 1/2 2 2 1/2
csgn(x + I y) ln(1/2 (x + y + 2 y + 1) + 1/2 (x + y - 2 y + 1)
2 2 1/2 2 2 1/2 2 1/2
+ ((1/2 (x + y + 2 y + 1) + 1/2 (x + y - 2 y + 1) ) - 1) )
+ i
2 2 1/2 2 2 1/2
arcsin(1/2 (x + y + 2 y + 1) - 1/2 (x + y - 2 y + 1) )
ARCCOSH
z+i z-i
arccosh z = 2 ln( sqrt --- + sqrt --- )
2 2
arccosh z = ln(z + sqrt(z-1) sqrt(z+1) ) not sqrt(z**2-1)
1 3 15
arccosh z = ln(2z) - --- - ---- - ----- - ... for |z| > 1
2 4 6
4z 32z 288z
arccosh(cosh(z)) = z
arccosh z =
2 2 1/2
- csgn(I - I x + y) csgn(I x - y) ln(1/2 (x + 2 x + 1 + y )
2 2 1/2
+ 1/2 (x - 2 x + 1 + y )
2 2 1/2 2 2 1/2 2 1/2
+ ((1/2 (x + 2 x + 1 + y ) + 1/2 (x - 2 x + 1 + y ) ) - 1) )
+ i
csgn(I - I x + y)
2 2 1/2 2 2 1/2
arccos(1/2 (x + 2 x + 1 + y ) - 1/2 (x - 2 x + 1 + y ) )
ARCTANH
arctanh z = ( ln(1+z) - ln(1-z) )/2
1 1+z
arctanh z = - ln --- must fix up on slit for z > 1
2 1-z
3 5 7
z z z
arctanh z = z + -- + -- + -- + ... for |z| < 1
3 5 7
arctanh z = -i arctan iz
arctanh(tanh(z)) = z
arctanh z = arccoth(z) + i Pi/2
2 2
(x + 1) + y
arctanh z = 1/4 ln(-------------)
2 2
(x - 1) + y
+i 1/2 arctan(y, x + 1) - 1/2 arctan(- y, 1 - x)
ARCCOTH
1 z+1
arccoth z = - ln --- must fix up on slit
2 z-1
3 5
i Pi z z
arccoth z = ---- + z + -- + -- + ... for |z| < 1
2 3 5
arccoth z = arctanh(z) +i Pi/2
arccoth z = arctanh(1/z)
arccoth(coth(z)) = z
2 2
(x + 1) + y
arccoth z = 1/4 ln(-------------)
2 2
(x - 1) + y
+i 1/2 Pi + 1/2 arctan(y, x + 1) - 1/2 arctan(- y, 1 - x)
Range:
Real part mathematically unbounded, imaginary part in [0.0 , i Pi]
There are many complex functions provided by Ada
generic_complex_elementary_functions.ads
Fortran 90 has a few complex functions
complex_func.f90 source code
complex_func_f90.out output
Python has many complex functions
test_complex.py source code
test_complex_py.out output
Haskell has many complex functions
test_complex.hs source code
test_complex_hs.out output
Complex.hs Library module
I programmed some complex functions in java.
Complex.java source code
MatLab overloads all functions that take floating point
to also handle complex numbers.
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