fix cosmetic issues

src/1_complex_numbers.md

@@ -159,17 +159,13 @@ 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:
+obtain the *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 $z$. ]{}
+differentiable at the point $z$.
 
 Note that differentiability is a property which not only pertains to a
 function, but also to a point.
@@ -287,8 +283,6 @@ $$\tanh'(x) = 1 + \frac{\sinh^2 x}{\cosh^2 (x)} = - \frac{1}{\cosh^2(x)}.$$
 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
@@ -336,22 +330,21 @@ TODO: Here was the beginning of a mdframed env
     -   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
 ========
 
-1.  [\[]{}D1[\]]{} Given $a=1+2\rm i$ and $b=4-2\rm i$, draw in the
+1.  *(difficulty: +)*  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
     $\ln(a)$.
 
-2.  [\[]{}D1[\]]{} Evaluate (i) $\rm i^{1/4}$, (ii)
+2.  *(difficulty: +)* Evaluate (i) $\rm i^{1/4}$, (ii)
     $\left(-1+\rm i \sqrt{3}\right)^{1/2}$, (iii) $\exp(2\rm i^3)$.
 
-3.  [\[]{}D1[\]]{} Evaluate $$\left| \frac{a+b\rm i}{a-b\rm i} \right|$$
+3.  *(difficulty: +)* Evaluate $$\left| \frac{a+b\rm i}{a-b\rm i} \right|$$
     for real $a$ and $b$.
 
-4.  [\[]{}D1[\]]{} Show that $\cos x = \cosh(\rm i x)$ and
+4.  *(difficulty: +)* Show that $\cos x = \cosh(\rm i x)$ and
     $\cos(\rm i x) = \cosh x$. Derive similar relations for $\sinh$ and
     $\sin$.
 
@@ -360,28 +353,28 @@ Problems
     Also show that $\cosh x$ is a solution to the differential equation
     $$y'' = \sqrt{1 + y'^2}.$$
 
-5.  [\[]{}D1[\]]{} Calculate the real part of
+5.  *(difficulty: +)* Calculate the real part of
     $\int_0^\infty e^{-\gamma t  +\rm i \omega t} dt$ ($\omega$ and
     $\gamma$ are real; $\gamma$ is positive).
 
-6.  [\[]{}D1[\]]{} Is the function $f(z) = |z| = \sqrt{x^2 + y^2}$
+6.  *(difficulty: +)* 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?
 
-7.  [\[]{}D1[\]]{} Show that the Cauchy-Riemann equations imply that the
+7.  *(difficulty: +)* 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 $v$.
 
-8.  [\[]{}D3[\]]{} Show that the set of points $z$ obeying
+8.  *(difficulty: +++)* 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 $\lambda = 1$?
 
-9.  [\[]{}D2[\]]{} In two dimensions, the Coulomb potential is
+9.  *(difficulty: ++)* 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 $z$
@@ -397,7 +390,7 @@ Problems
     This is called a *multipole expansion*. A similar expansion exist in
     three dimensions.
 
-10. [\[]{}D2[\]]{} In this problem, we consider the function $1/z$ close
+10. *(difficulty: ++)* 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