Update src/1_complex_numbers.md

src/1_complex_numbers.md

@@ -17,7 +17,8 @@ The lecture on complex numbers consists of three parts, each with their own vide
 <iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/fLMdaMuEp8s?rel=0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
 
 ### Complex numbers 
-Complex numbers are numbers of the form $$z = a + b {\rm i}.$$ 
+!!! info "Definition"
+    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 on numbers have their natural extension for complex
@@ -37,33 +38,25 @@ For two complex numbers, $z_1 = a_1 + b_1 {{\rm i}}$ and $z_2 = a_2 + b_2 {{\rm
 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:
+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 $\rm i$).
+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
+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.
+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 $z$:
+$$z z^* = (a+b{{\rm i}})(a-b{{\rm i}}) = a^2 + b^2.$$ 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 $z_2$:
+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!
 
@@ -99,7 +92,7 @@ by the two numbers.
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-Real functions can (most of the times) be written in terms of a Taylor series:
+Real functions can (most of the times) be written in terms of a Taylor series at a point $x_{0}$:
 $$f(x) = \sum \limits_{n=0}^{\infty} \frac{f^{(n)}(x_{0})}{n!} (x-x_{0})^{n}$$
 We can write something similar for complex functions, 
 when replacing the *real* variable $x$ with its *complex* counterpart $z$:
@@ -109,12 +102,12 @@ For this course, the most important function is the *complex exponential functio
 
 ### 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.
+It occurs in Fourier transforms and it is very convenient for doing calculations involving cosines and sines. 
+It also makes performing many common operations on complex number a lot easier.
 
-The exponential function $f(z) = \exp(z) = e^z$ is defined as:
-$$\exp(z) = e^{x + {\rm i}y} = e^{x} e^{{\rm i} y} = e^{x} \left( \cos y + {\rm i} \sin y\right).$$
+!!! info "The exponential function and Euler identity"
+    The exponential function $f(z) = \exp(z) = e^z$ is defined as:
+    $$\exp(z) = e^{x + {\rm i}y} = e^{x} e^{{\rm i} y} = e^{x} \left( \cos y + {\rm i} \sin y\right).$$
 The last expression is called the *Euler identity*.
 
 **Exercise** Check that this function obeys