1_complex_numbers.md



title: Complex Numbers


Complex numbers
Complex numbers are numbers of the form 
z = a + b {\rm i}. Here
\rm i
 is the square root of -1: {\rm i} = \sqrt{-1}, or,
equivalently: {\rm i}^2 = -1.

Usual operations for numbers have their natural extension for complex
numbers as we shall see below.
Some definitions:


For a complex number 
z = a + b {{\rm i}}
, a
 is called the real
part, and b
 the imaginary part.


The complex conjugate 
z^*
 of z = a + b {{\rm i}} is defined as
z^* = a - b{{\rm i}}, i.e., taking the complex conjugate means
flipping the sign of the imaginary part.


Addition
For two complex numbers, 
z_1 = a_1 + b_1 {{\rm i}} and
z_2 = a_2 + b_2 {{\rm i}}
, the sum w = z_1 + z_2 is given as
w = w_1 + w_2 {{\rm i}}= (a_1 + a_2) + (b_1 + b_2) {{\rm i}} where
the parentheses in the rightmost expression have been added to group the
real and the imaginary part. A consequence of this definition is that
the sum of a complex number and its complex conjugate is real:
z + z^* = a + b {{\rm i}}+ a - b {{\rm i}}= 2a, i.e., this results
in twice the real part of z
. Similarly, subtracting z^*
 from z
yields z - z^* = a + b {{\rm i}} - a + b {{\rm i}}= 2b{\rm i}, i.e.,
twice the imaginary part of z
 (times \rm i
).

Multiplication
For the same two complex numbers 
z_1
 and z_2 as above, their product
is calculated as
w = z_1 z_2 = (a_1 + b_1 {{\rm i}}) (a_2 + b_2 {{\rm i}}) = (a_1 a_2 - b_1 b_2) + (a_1 b_2 + a_2 b_1) {{\rm i}},
where the parentheses have again be used to indicate the real and
imaginary parts.
A consequence of this definition is that the product of a complex number
z = a + b {{\rm i}} with its conjugate is real:
z z^* = (a+b{{\rm i}})(a-b{{\rm i}}) = a^2 + b^2. The square root of
this number is the norm |z|
 of z:
|z| = \sqrt{z z^*} = \sqrt{a^2 + b^2}.


Division
The quotient 
z_1/z_2
 of two complex numbers z_1
 and z_2 as above,
can be evaluated by multiplying the numerator and denominator by the
complex conjugate of z_2:
\frac{z_1}{z_2} = \frac{z_1 z_2^*}{z_2 z_2^*} = \frac{(a_1 a_2 + b_1 b_2) + (-a_1 b_2 + a_2 b_1) {{\rm i}}}{a_2^2 + b_2^2}.
Check this!

The complex plane
Complex numbers can be rendered on a two-dimensional (2D) plane, the
complex plane. This plane is spanned by two unit vectors, one
horizontal, which represents the real number 1, whereas the vertical
unit vector represents 
{\rm i}
.

[

]{}
Note that the norm of 
z
 is the length of this vector.

Addition in the complex plane
Adding two numbers in the complex plane corresponds to adding the
horizontal and vertical components:

[

]{}
We see that the sum is found as the diagonal of a parallelogram spanned
by the two numbers.

Argument and Norm
A complex number can be represented by two real numbers, 
a
 and b
which represent the real and imaginary part of the complex number. An
alternative representation is a vector in the complex plane, whose
horizontal component is the real, and vertical component the imaginary
part. However, it is also possible to characterize that vector by its
length and direction, where the latter can be represented by the
angle the vector makes with the horizontal axis:

[

]{}
The angle with the horizontal axis is denoted by 
\varphi, just as in
the case of polar coordinates. In the context of complex numbers, this
angle is denoted as the argument. We have:

A complex number can be represented either by its real and imaginary
part, corresponding to the Cartesian coordinates in the complex plane,
or by its norm and its argument, corresponding to polar
coordinates. The norm is the length of the vector, and the argument is
the angle it makes with the horizontal axis.

From our previous discussion on polar coordinates we can conclude that
for a complex number 
z = a + b {\rm i}, its real and imaginary parts
can be expressed as a = |z| \cos\varphi
 b = |z| \sin\varphi The
inverse equations are |z| = \sqrt{a^2 + b^2}
\varphi = \arctan(b/a)
 for a>0. In general:
\varphi = \begin{cases} \arctan(b/a) &{\rm for ~} a>0; \\
\pi + \arctan(b/a) & {\rm for ~} a<0 {\rm ~ and ~} b>0;\\
-\pi + \arctan(b/a) &{\rm ~ for ~} a<0 {\rm ~ and ~} b<0.
\end{cases}


Complex functions
A complex function 
f
 maps any complex number z onto another complex
number f(z). Just as with real functions, we can define the derivative
of a complex function:
\frac{df(z)}{dz} = \lim_{dz \rightarrow 0} \frac{f(z+dz) - f(z)}{dz}.
It is useful to contemplate this expression for some time. The number
dz in this expression is a complex number with vanishingly small
norm. However, its direction (i.e. its argument) is not specified. A
complex function is called differentiable only if the limit exists and
gives one and the same number, irrespective of the direction of dz.
Let’s see whether we can find a condition for this to happen. We assume
that the function f is continuous. Note that, as f is complex, it
can be written as a real plus an imaginary part:
f(z) = u(z) + {\rm i} v(z). For z=x + \rm i y, we can write the
two functions u and v in terms of x and y. Let us assume that
the two partial derivatives of these functions with respect to x and
y exist.
Calculating the partial derivatives of f with respect to x and y
yields:
\frac{\partial f(z)}{\partial x} = \lim_{dx \rightarrow 0} \frac{f(z+dx) - f(z)}{dx} = \frac{\partial u}{\partial x} + {\rm i} \frac{\partial v}{\partial x},
and
\frac{\partial f(z)}{\partial y} = \lim_{dy \rightarrow 0} \frac{f(z+{\rm i} dy) - f(z)}{dy} = \frac{\partial u}{\partial y} + {\rm i} \frac{\partial v}{\partial y}.
Unlike df/dz, these derivatives are always defined. In terms of these
two partial derivatives, we can find the derivative with respect to any
direction using first-order Taylor expansions. Taking
dz = d x + {\rm i} dy, we have
f(z + dz) = u(x+dx, y+dy) + {\rm i} v(x+dx, y+dy) = u(x,y) + {\rm i} v(x,y) + dx \frac{\partial u}{\partial x} + dy \frac{\partial u}{\partial y} + 
{\rm i} \left[ dx \frac{\partial v}{\partial x} + dy \frac{\partial  v}{\partial y} \right].
Collecting the terms proportional to dx and dy respectively we
obtain:
f(z + dz ) = f(z) + \left( \frac{\partial u}{\partial x} + {\rm i} \frac{\partial v}{\partial x} \right) dx +
\left( \frac{\partial u}{\partial y} + {\rm i} \frac{\partial v}{\partial y} \right) dy.
Let’s write
f(z + dz ) - f(z) = \left( \frac{\partial u}{\partial x} + {\rm i}\frac{\partial v}{\partial x} \right) dx +
{\rm i} \left( -{\rm i} \frac{\partial u}{\partial y} + \frac{\partial v}{\partial y} \right) dy.
If the derivative df/dz exists, the right hand side divided by dz
should be independent of dz=dx + {\rm i} dy! Thus, f(z) is
differentiable only when
\frac{\partial u}{\partial x} + {\rm i} \frac{\partial v}{\partial x} = -{\rm i} \frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}.
Equating the real and imaginary parts of the left and right hand side we
obtain the
TODO: Here was a remark environment
[ Cauchy Riemann differential equations:
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} {~~~ \rm and ~~~ } \frac{\partial v}{\partial x} = - 
\frac{\partial u}{\partial y}. The derivative is then given as
\frac{df}{dz} = \frac{\partial u}{\partial x} + {\rm i} \frac{\partial v}{\partial x}.
A complex function whose real and imaginary part (u and v) obey the
Cauchy-Riemann differential equations in a point z, is complex
differentiable at the point z. ]{}
Note that differentiability is a property which not only pertains to a
function, but also to a point.
It can be shown that a complex function which is differentiable in z,
is infinitely differentiable. Such a function is called analytic in
the point z. An analytic function can be expanded as an inifinite
series near any point where the function is analytic. Here, “near any”
point indicates that the series converges and coincides with f only on
a circle in the complex plane. The radius of that circle is limited to
the distance of z to the nearest point w where the function is not
analytic. This distance is called the convergence radius.
Recall that the Taylor series around a complex point a has the form
f(z) = \sum_{j=0}^\infty d_j (z-a)^j, where
d_j = \frac{1}{j!} f^{(j)} (a). Here, f^{(j)}(a) denotes the
j-th derivative of the function f in the point a.

The complex exponential function
The exponential function f(z) = \exp(z) = e^z is defined as:
\exp(z) = e^{x} \left( \cos y + {\rm i} \sin y\right).
Exercise Check that this function obeys
\exp(z_1) \exp(z_2) = \exp(z_1 + z_2). You need sum- and difference
formulas of cosine and sine.
Exercise Check that \exp(z) obeys the Cauchy-Riemann equations and
that the derivative is the exponential function itself:
\frac{d \exp(z)}{dz} = \exp(z). Note that, for any complex number
z, we can write z = |z| e^{\rm i \varphi}, where
\varphi = \text{arg}(z).
In real calculus, the logarithmic function is the inverse of the
exponential function. Similarly, we want the complex logarithm to be the
inverse of the complex exponential function. Let’s write
w = \exp(z) = e^x(\cos y + \rm i \sin y). We know then that
\log(w) = z = x + \rm i y. Realising that the norm of
\cos y + \rm i \sin y is 1 (check this!), we see that
\left|w\right| = e^x. Therefore, the real part of \log w is the real
logarithm of |w|.
The imaginary part of the \log w should be y. Now, y is the
argument of w. All in all, we therefore see that
\log w = \log|w| + \rm i \arg(w).
The complex exponential is used extremely often. It occurs in Fourier
transforms and is very convenient for doing calculations involving
cosines and sines.
Let us show some tricks where the simple properties of the exponential
function helps in re-deriving trigonometric identities.


Take |z_1| = |z_2| = 1, and \arg{(z_1)} = \varphi_1 and
\arg{(z_2)} = \varphi_2. Then it is easy to see that
z_i = \exp({\rm i} \varphi_i), i=1, 2. Then:
z_1 z_2 = \exp[{\rm i} (\varphi_1 + \varphi_2)]. The left hand
side can be written as
z_1 z_2 = \left[ \cos(\varphi_1) + {\rm i} \sin(\varphi_1) \right] \left[ \cos(\varphi_2) + {\rm i} \sin(\varphi_2) \right] = \cos\varphi_1 \cos\varphi_2 - \sin\varphi_1 \sin\varphi_2 + {\rm i} \left( \cos\varphi_1 \sin\varphi_2 + 
\sin\varphi_1 \cos\varphi_2 \right). On the other hand, the right
hand side can be written as
\exp[{\rm i} (\varphi_1 + \varphi_2)] = \cos(\varphi_1 + \varphi_2) + {\rm i} \sin(\varphi_1 + \varphi_2).
Comparing the two expressions, equating their real and imaginary
parts, we find
\cos(\varphi_1 + \varphi_2) = \cos\varphi_1 \cos\varphi_2 - \sin\varphi_1 \sin\varphi_2;
\sin(\varphi_1 + \varphi_2) = \cos\varphi_1 \sin\varphi_2 + 
\sin\varphi_1 \cos\varphi_2. Note that we used the resulting
formulas already in order to derive the properties of the
exponential function. The point is that you can use the properties
of the complex exponential to quickly find the form of gonometric
formulas which you easily forget.


As a final example, consider what we can learn from the derivative
of the exponential function:
\frac{d}{d\varphi} \exp({\rm i} \varphi) = {\rm i} \exp({\rm i} \varphi) .
Writing out the exponential in terms of cosine and sine, we see that
\cos'\varphi + {\rm i} \sin'\varphi = {\rm i} \cos\varphi - \sin\varphi.
where the prime ' denotes the derivative as usual. Equating real
and imaginary parts leads to \cos'\varphi = - \sin\varphi;
\sin'\varphi = \cos\varphi.


Hyperbolic functions
From
e^{\rm i \varphi} = \left( \cos\varphi + {\rm i} \sin\varphi\right),
it immediately follows that
\cos\varphi = \frac{e^{{\rm i} \varphi} + e^{-{\rm i} \varphi}}{2}.
and
\sin\varphi = \frac{e^{{\rm i} \varphi} - e^{-{\rm i} \varphi}}{2{\rm i}}.
It is then tempting to generalise these functions for imaginary angles.
These functions are known as hyperbolic functions. They are are called
the hyperbolic cosine and hyperbolic sine functions and they are denoted
as \sinh and \cosh: \cosh(x) = \frac{e^x + e^{-x}}{2};
\sinh(x) = \frac{e^x - e^{-x}}{2}. From these definitions the
following properties can easily be derived.


Derivatives \frac{d\cosh(x)}{dx} = \sinh(x);
\frac{d\sinh(x)}{dx} = \cosh(x).


\cosh^2(x) - \sinh^2(x) = 1.


‘Double angle’ formulas: \cosh(2x) = \cosh^2(x) + \sinh^2(x);
\sinh(2x) = 2\cosh(x) \sinh(x).


It may seem that these function are rather exotic; however they occur in
everyday life: the shapes of power lines and of soap films can be
described by hyperbolic cosines and sines!
Finally, the hyperbolic tangent is defined as
\tanh(x) = \frac{\sinh(x)}{\cosh(x)}. Its derivative is given as
\tanh'(x) = 1 + \frac{\sinh^2 x}{\cosh^2 (x)} = - \frac{1}{\cosh^2(x)}.

Summary
TODO: Here was the beginning of a mdframed env


A complex number z has the form z = a + b \rm i where a and
b are both real, and \rm i^2 = 1. The real number a is called
the real part of z and b is the imaginary part. Two complex
numbers can be added, subtracted and multiplied straightforwardly.
The quotient of two complex numbers z_1=a_1 + \rm i b_1 and
z_2=a_2 + \rm i b_2 is
\frac{z_1}{z_2} = \frac{z_1 z_2^*}{z_2 z_2^*} = \frac{(a_1 a_2 + b_1 b_2) + (-a_1 b_2 + a_2 b_1) {{\rm i}}}{a_2^2 + b_2^2}.


Complex numbers can also be characterised by their norm
|z|=\sqrt{a^2+b^2} and argument \phi. These coordinates
correspond to polar coordinates in the complex plane. For a complex
number z = a + b {\rm i}, its real and imaginary parts can be
expressed as a = |z| \cos\varphi b = |z| \sin\varphi The
inverse equations are |z| = \sqrt{a^2 + b^2}
\varphi = \begin{cases} \arctan(b/a) &{\rm for ~} a>0; \\
\pi + \arctan(b/a) & {\rm for ~} a<0 {\rm ~ and ~} b>0;\\
-\pi + \arctan(b/a) &{\rm ~ for ~} a<0 {\rm ~ and ~} b<0.
\end{cases}


The derivative of a complex function f(z) is defined as
\frac{df(z)}{dz} = \lim_{dz \rightarrow 0} \frac{f(z+dz) - f(z)}{dz}.
The right hand side depends on the direction of dz in the complex
plane. The function is said to be differentiable if the right hand
side gives a unique value. This is the case when the real part u
and imaginary part v of the function f satisfy the
Cauchy–Riemann equations:
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} {~~~ \rm and ~~~ } \frac{\partial v}{\partial x} = - 
\frac{\partial u}{\partial y}. A function which is differentiable,
is differentiable infinitely often. Such a function can be expanded
as a Taylor series:
f(z) = \sum_{j=0}^\infty \frac{1}{j!} f^{(j)} (a) (z-a)^j, where
f^{(j)} (a) is the j-th derivative of the function f in a.


Examples of differentiable functions:


The complex exponential:
e^z = e^{x} \left( \cos y + \rm i \sin y\right).


The complex logarithm: \log(z) = \log|z| +\rm i \arg(z).


The complex sine and cosine functions are defined as
\sin(z) = \frac{e^{\rm i z} - e^{-\rm i z}}{2\rm i}; \phantom{xxx} \cos(z) = \frac{e^{\rm i z} + e^{-\rm i z}}{2}.
The complex tangent is defined as \tan(z) = \sin(z)/\cos(z).


Hyperbolic functions are defined as:
\sinh(z) = \frac{e^{z} - e^{-z}}{2}; \phantom{xxx} \cosh(z) = \frac{e^{z} + e^{-z}}{2}.


TODO: Here was the end of a mdframed env

Problems


[[]{}D1[]]{} Given a=1+2\rm i and b=4-2\rm i, draw in the
complex plane the numbers a+b, a-b, ab, a/b, e^a and
\ln(a).


[[]{}D1[]]{} Evaluate (i) \rm i^{1/4}, (ii)
\left(-1+\rm i \sqrt{3}\right)^{1/2}, (iii) \exp(2\rm i^3).


[[]{}D1[]]{} Evaluate \left| \frac{a+b\rm i}{a-b\rm i} \right|
for real a and b.


[[]{}D1[]]{} Show that \cos x = \cosh(\rm i x) and
\cos(\rm i x) = \cosh x. Derive similar relations for \sinh and
\sin.
Show that \cosh^2 x - \sinh^2 x = 1.
Also show that \cosh x is a solution to the differential equation
y'' = \sqrt{1 + y'^2}.


[[]{}D1[]]{} Calculate the real part of
\int_0^\infty e^{-\gamma t  +\rm i \omega t} dt (\omega and
\gamma are real; \gamma is positive).


[[]{}D1[]]{} Is the function f(z) = |z| = \sqrt{x^2 + y^2}
analytic on the complex plane or not? If not, where is the function
not analytic?


[[]{}D1[]]{} Show that the Cauchy-Riemann equations imply that the
real and imaginary part of a differentiable complex function both
represent solutions to the Laplace equation, i.e.
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0,
for the real part u of the function, and similarly for the
imaginary part v.


[[]{}D3[]]{} Show that the set of points z obeying
| z - \rm i a| = \lambda |z + \rm i a|, with a and \lambda
real, form a circle with radius 2|\lambda/(1-\lambda^2) a|
centered on the point \rm i a (1+\lambda^2)/(1-\lambda^2),
provided \lambda \neq 1. What is the set like for \lambda = 1?


[[]{}D2[]]{} In two dimensions, the Coulomb potential is
proportional to \log |r|. Viewing the 2D space as a complex plane,
this is \log |z|. Consider a system consisting of charges q_i
placed at ‘positions’ z_i, all close to the origin. The point z
is however located far away from the origin. Use the Taylor
expansions of the terms \ln(z-z_i) at z with respect to the
z_i to write the potential at z as
U(z) = \sum_i q_i\ln(z-z_i) = a_0 \ln z - \sum_{k=1}^M \frac{a_k}{z^k} + {\mathcal O} \left( \frac{R}{z} \right)^{M+1}
where
a_0 = \sum_i q_i {\rm ~~ and ~~} a_k = \sum_{i=1}^{N_c} \frac{q_i z_i^k}{k}, k\geq 1.
(Note that, for the potential, the real part of the right hand side
is to be taken.)
This is called a multipole expansion. A similar expansion exist in
three dimensions.


[[]{}D2[]]{} In this problem, we consider the function 1/z close
to the real axis: z=x-\rm i \epsilon where \epsilon is small.
Show that the imaginary part of this function approaches \pi times
the Dirac delta-function \delta(x) for \epsilon\rightarrow 0. Do
this by showing that the integral over that function is \pi and
that, when multiplied by a function f(x), the result only depends
on f(0).