Updated: complex functions, differentiation/integration

Added: usefull properties and relations.
Removed: Complex functions in summary, 

src/1_complex_numbers.md

@@ -110,42 +110,53 @@ inverse equations are $$|z| = \sqrt{a^2 + b^2}$$
 $$\varphi = \arctan(b/a)$$ for $a>0$. In general:
 $$\varphi = \begin{cases} \arctan(b/a) &{\rm for ~} a>0; \\
  \pi + \arctan(b/a) & {\rm for ~} a<0 {\rm ~ and ~} b>0;\\
- -\pi + \arctan(b/a) &{\rm ~ for ~} a<0 {\rm ~ and ~} b<0.
+ -\pi + \arctan(b/a) &{\rm for ~} a<0 {\rm ~ and ~} b<0.
  \end{cases}$$
 
+## Complex functions
+Real functions can (most of the times) be written in terms of a Taylor series:
+$$f(x) = \sum \limits_{n=0}^{\infty} \frac{f^{(n)}(x_{0})}{n!} (x-x_{0})^{n}$$
+We can write something similar for complex functions, 
+when replacing $x \lim z$:
+$$f(z) = \sum \limits_{n=0}^{\infty} \frac{f^{(n)}(x_{0})}{n!} (z-x_{0})^{n}$$
 
+We have already seen that we can write any complex number in polar form, with a normal and complex exponential.
+The most important complex function for us then also is the complex exponential function, at which we will have a look below.
 
 ## The complex exponential function
+The complex exponential is used *extremely often*. 
+It occurs in Fourier transforms and it is very convenient for doing calculations 
+involving cosines and sines. 
+It also makes doing many common operations on complex number a lot easier.
 
 The exponential function $f(z) = \exp(z) = e^z$ is defined as:
-$$\exp(z) = e^{x} \left( \cos y + {\rm i} \sin y\right).$$
+$$\exp(z) = e^{x + {\rm i}y} = e^{x} + e^{{\rm i} y} = e^{x} \left( \cos y + {\rm i} \sin y\right).$$
+The last expression is called the *Euler identity*.
 
 **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.
+Some useful values of the complex exponential to know by heart are $e^{2{\rm i } \pi} = 1 $, $e^{{\rm i} \pi} = -1 $ and $e^{{\rm i} \pi/2} = {\rm i}$. 
+From the first expression, it also follows that 
+$$e^{{\rm i} (y + 2\pi n)} = e^{{\rm i}\pi} {rm ~ for ~} n \in \Z$$
+As a result, $y$ is only defined up to $2\pi$.
+
+Furthermore, we can define the sine and cosine in terms of complex exponentials:
+$$\cos(x) = \frac{e^{{\rm i} x} + e^{-{\rm i} x}}{2}$$
+$$\sin(x) = \frac{e^{{\rm i} x} - e^{-{\rm i} x}}{2}$$
+
+
+$$z^{n} = \left(r e^{{\rm i} \phi}\right)^{n} = r^{n} e^{{\rm i} n \phi}$$
+$$\sqrt[n]{z} = \sqrt[n]{r e^{{\rm i} \phi} } = \sqrt[n]{r} e^{{\rm i}\phi/n} $$
+$$\log(z) = log \left(r e^{{\rm i} \phi}\right) = log(r) + {\rm i} \phi$$
+$$z_{1}z_{2} = r_{1} e^{{\rm i} \phi_{1}} r_{2} e^{{\rm i} \phi_{2}} = r_{1} r_{2} e^{{\rm i} (\phi_{1} + \phi_{2}}$$
+
+### Differentiation and integration
+We only consider differentiation and integration over *real* variables. We can then regard the complex ${\rm i}$ as another constant, and use our usual differentiation and integration rules:
+$$\frac{d}{d\varphi} e^{{\rm i} \varphi} = e^{{\rm i} \varphi} \frac{d}{d\varphi} ({\rm i} \varphi) ={\rm i} e^{{\rm i} \varphi} .$$
+$$\int_{0}^{\pi} e^{{\rm i} \varphi} = \frac{1}{{\rm i}} \left[ e^{{\rm i} \varphi} \right]_{0}^{\pi} = -{\rm i}(-1 -1) = 2 {\rm i}$$
+
 
 Let us show some tricks where the simple properties of the exponential
 function helps in re-deriving trigonometric identities.
@@ -212,7 +223,6 @@ $$\tanh'(x) = 1 + \frac{\sinh^2 x}{\cosh^2 (x)} = - \frac{1}{\cosh^2(x)}.$$
 
 ## Summary
 
-
 -   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
@@ -232,35 +242,6 @@ $$\tanh'(x) = 1 + \frac{\sinh^2 x}{\cosh^2 (x)} = - \frac{1}{\cosh^2(x)}.$$
      -\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:
-        $$\sinh(z) = \frac{e^{z} - e^{-z}}{2}; \phantom{xxx} \cosh(z) = \frac{e^{z} + e^{-z}}{2}.$$
-
-
 ## Problems
 
 1.  [:grinning:]  Given $a=1+2\rm i$ and $b=4-2\rm i$, draw in the