diff --git a/src/1_complex_numbers.md b/src/1_complex_numbers.md index e55911b8e26c04924487091a9624676d5925c2d4..3b203cb2dda0c1ccfdca062743043021cb253eb3 100644 --- a/src/1_complex_numbers.md +++ b/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