diff --git a/src/1_complex_numbers.md b/src/1_complex_numbers.md index ea01b4f70ae8fb969601c38f458ced421ba45ed7..60abfdc0d025242c08e00de46cda96125266a085 100644 --- a/src/1_complex_numbers.md +++ b/src/1_complex_numbers.md @@ -287,27 +287,33 @@ function help in re-deriving trigonometric identities. ## 1.6. Problems -1. [:grinning:] Given $a=1+2\rm i$ and $b=-3+4\rm i$, calculate and draw in the - complex plane the numbers $a+b$, $ab$, and $b/a$. - -2. [:grinning:] Evaluate (a) $\rm i^{1/4}$, (b) - $\left(1+\rm i \sqrt{3}\right)^{1/2}$, (c) $\exp(2\rm i^3)$. - -3. [:grinning:] Find the three 3rd roots of $1$ and ${\rm i}$ (i.e. all possible solutions to the equations $x^3 = 1$ and $x^3 = {\rm i}$, respectively). - -4. [:grinning:] (a) Find the real and imaginary part of - $$ \frac{1+ {\rm i}}{2+3{\rm i}}$$ - (b) Evaluate $$\left| \frac{a+b\rm i}{a-b\rm i} \right|$$ - for real $a$ and $b$. - -5. [:sweat:] For any given complex number $z$, we can take the inverse $\frac{1}{z}$. Visualize taking the inverse in the complex plane. What geometric operation does taking the inverse correspond to? (Hint: first consider what geometric operation $\frac{1}{z^*}$ corresponds to.) - -6. [:grinning:] Compute (a) - $$\frac{d}{dt} e^{{\rm i} (kx-\omega t)},$$ - and (b) calculate the real part of - $$\int_0^\infty e^{-\gamma t +\rm i \omega t} dt$$($k$, $x$, $\omega$, $t$ and - $\gamma$ are real; $\gamma$ is positive). - -7. [:smirk:] Compute +1. [:grinning:] Given $a=1+2\rm i$ and $b=-3+4\rm i$, calculate and draw in the complex plane the numbers: + 1. $a+b$, + 2. $ab$, + 3. $b/a$. + +2. [:grinning:] Evaluate: + 1. $\rm i^{1/4}$, + 2. $\left(1+\rm i \sqrt{3}\right)^{1/2}$, + 3. $\exp(2\rm i^3)$. + +3. [:grinning:] Find the three 3rd roots of $1$ and ${\rm i}$. </br> + (i.e. all possible solutions to the equations $x^3 = 1$ and $x^3 = {\rm i}$, respectively). + +4. [:grinning:] *Quotients*</br> + 1. Find the real and imaginary part of $$ \frac{1+ {\rm i}}{2+3{\rm i}} \, .$$ + 2. Evaluate for real $a$ and $b$:$$\left| \frac{a+b\rm i}{a-b\rm i} \right| \, .$$ + +5. [:sweat:] For any given complex number $z$, we can take the inverse $\frac{1}{z}$. + 1. Visualize taking the inverse in the complex plane. + 2. What geometric operation does taking the inverse correspond to? </br> + (Hint: first consider what geometric operation $\frac{1}{z^*}$ corresponds to.) + +6. [:grinning:] *Differentation and integration* </br> + 1. Compute $$\frac{d}{dt} e^{{\rm i} (kx-\omega t)},$$ + 2. Calculate the real part of $$\int_0^\infty e^{-\gamma t +\rm i \omega t} dt$$ + ($k$, $x$, $\omega$, $t$ and $\gamma$ are real; $\gamma$ is positive). + +7. [:smirk:] Compute by making use of the Euler identity. $$\int_{0}^{\pi}\cos(x)\sin(2x)dx$$ - by making use of the Euler identity. +