Some Elementary Functions of Complex Variable z
The following elementary functions are of importance in dealing with complex variables:
Exponential function: Using the basic definition of exponential function for real
variables, one gets
∞ n
z
z 2 z3 z 4
+
+
+K
f(z) = ez = ∑ = 1 + z +
n
!
2
!
3
!
4
!
k =1
with
u(x,y) = ex cosy & v(x,y) = ex siny
and
mod(ez) = |ez| = ex & arg(ez) = y
Note that
∞
(− z) n
z 2 z3 z 4
e =∑
=1− z + − + +K
2! 3! 4!
k =1 n!
–z
∞
iz
e =
(iz) n
z 2 iz3 z 4
1
iz
=
+
−
−
+ +K
∑ n!
2
!
3
!
4!
k =1
∞
–iz
e
(−iz) n
z 2 iz3 z 4
=∑
= 1 − iz − +
+ +K
n
!
2
!
3
!
4!
k =1
Hyperbolic functions: Using ez and e–z expressions given above, one gets the following
expressions:
cosh(z) = (ez + e–z)/2 = cosh(x)cos(y) + i sinh(x)sin(y)
sinh(z) = (ez – e–z)/2 = sinh(x)cos(y) + i cosh(x)sin(y)
which we all familiar with for real variables.
If z = 0 + iy is used above, one finds
cosh(iy) = cos(y)
sinh(iy) = i sin(y)
Familiar laws for the hyperbolic functions of real quantities all hold for the hyperbolic
functions of complex quantities in the following sense:
cosh2(z) – sinh2(z) = 1
cosh(z1+z2) = cosh(z1)cosh(z2) + sinh(z1)sinh(z2)
sinh(z1+z2) = sinh(z1)cosh(z2) + cosh(z1)sinh(z2)
cosh(2z) = cosh2(z) + sinh2(z) = 1 + 2 sinh2(z) = 2 cosh2(z) – 1
sinh(2z) = 2 sinh(z) cosh(z)
Trigonometric functions: Using eiz and e–iz expressions given above, one gets the
following expressions:
cos(z) = (eiz + e–iz)/2 = cos(x)cosh(y) – i sin(x)sinh(y)
sin(z) = (eiz – e–iz)/2i = sin(x)cosh(y) + i cos(x)sinh(y)
2006Spring/Bülent E. Platin
ME 210 / Some Elementary Functions of Complex Variable z - 9
which we all familiar with for real variables. If z = 0 + iy is used above, one finds
cos(iy) = cosh(y)
sin(iy) = i sinh(y)
Familiar laws for the trigonometric functions of real quantities all hold for the
trigonometric functions of complex quantities in the following sense:
cos2(z) + sin2(z) = 1
cos(z1+z2) = cos(z1)cos(z2) – sin(z1)sin(z2)
sin(z1+z2) = sin(z1)cos(z2) + cos(z1)sin(z2)
cos(2z) = cos2(z) – sin2(z) = 1 – 2 sin2(z) = 2 cos2(z) – 1
sin(2z) = 2 sin(z) cos(z)
Logarithmic function: The logarithm of z that is defined implicitly as the function
w = ln(z)
which satisfies the equation
z = ew or r eiθ = eu + i v = eu eiv
Hence,
eu = r or u = ln(r), and v = θ
Thus,
w = u + i v = ln(r) + i θ = ln |z| + i arg(z)
If the principal argument of z is denoted by Arg(z), then this equation can be rewritten as
ln(z) = ln |z| + i [Arg(z) + 2kπ] k = 0, ±1, ±2, …
indicating that complex logarithmic function is infinitely multi–valued. For n = 0, the
part (branch) of the logarithmic function is called as the principal value.
Familiar laws for the logarithms of real quantities all hold for the logarithms of complex
quantities in the following sense:
ln(z1z2) = ln(z1) + ln (z2)
ln(z1/z2) = ln(z1) – ln (z2)
ln(zk) = k ln(z) k = 0, ±1, ±2, …
2006Spring/Bülent E. Platin
ME 210 / Some Elementary Functions of Complex Variable z -10