diff --git a/src/1_complex_numbers.md b/src/1_complex_numbers.md index e2458142fc1ebfad0f68838da5517c0693b36c97..60abfdc0d025242c08e00de46cda96125266a085 100644 --- a/src/1_complex_numbers.md +++ b/src/1_complex_numbers.md @@ -287,30 +287,29 @@ 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: +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 +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> +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:] </br> - 1. Find the real and imaginary part of $$ \frac{1+ {\rm i}}{2+3{\rm i}}$$ - 2. Evaluate $$\left| \frac{a+b\rm i}{a-b\rm i} \right|$$ for real $a$ and $b$. +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:] </br> - 1. For any given complex number $z$, we can take the inverse $\frac{1}{z}$. - 2. Visualize taking the inverse in the complex plane. - 3. What geometric operation does taking the inverse correspond to? +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:] </br> +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).