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diff --git a/Problem Set 1.page b/Problem Set 1.page index 9ea5399..cf96a80 100644 --- a/Problem Set 1.page +++ b/Problem Set 1.page @@ -1,37 +1,29 @@ -## Countability +1. Write the formula for multiplication of complex numbers in rectangular coordinates. How does this relate to the ``angle sum'' formulae from trigonometry? -1. Group the following sets according to their cardinality: +- Use De Moivre's theorem to write down a ``triple angle'' formulae, i.e. closed form expressions for $\sin 3x$ and $\cos 3x$. - a. $\mathbb{N} = \{ 1,2,3,4,\dots \}$ - - $\mathbb{Z} = \{ \dots, -2, -1,0,1,2, \dots \}$ - - $\mathbb{N} \times \mathbb{N}$ - - $\mathbb{Q}$ = Set of all fractions $\frac{n}{m}$ where $n,m \in \mathbb{Z}$ - - $\mathbb{R}$ - - The open interval $(0,1)$ - - The closed interval $[0,1]$ - - $2^{\mathbb{N}}$ = Set of all subsets of $\mathbb{N}$. - - $2^{\mathbb{R}}$ = Set of all subsets of $\mathbb{R}$. - - $\mathbb{R}^{\mathbb{R}}$ = Set of all functions from $\mathbb{R}$ to itself. +- Show that every nonzero complex number has exactly $3$ cube roots. What are the cube roots of $i$? Draw them in the complex plane. -Cook up other examples and post them on the wiki! +- Show that the Cauchy-Riemann equations are equivalent to the following PDE: -2. Let $X$ be any set. Show that the cardinality of $2^{X}$ is larger than the cardinality of $X$. -(Hint: Let $f: X \to 2^X$ be a bijection. Consider the set of all elements $x \in X$ such that $x$ is not an element of $f(x)$.) +$df/dx + i df/dy = 0$ +You might want to use this fact in the problems below, though it's not necessary. -## Fourier Series +- Show that the function $f(z) = \overline{z}$ is not holomorphic, despite being angle-preserving. How does this function transform the complex plane? +- Show that the function $f(z) = z^n$ is holomorphic for any integer n (possibly negative!). How do these functions transform the complex plane? -1. Compute the Fourier Series of the following functions. Do both the exponential and sin/cos expansions. - a. $f(x) = \sin^3(3x)\cos^2(4x)$ - - $g(x) = x(x-2\pi)$ - (Hint: Use integration by parts) +- Show that the sum of two holomorphic functions is holomorphic; conclude that any polynomial function is holomorphic. -2. Show that -$\int_0^{2\pi} \sin^4(x) dx = \frac{3 \pi}{4}$ -(Hint: write out the exponential fourier expansion of $\sin^4(x)$.) +- Show that the product of two holomorphic functions is holomorphic. -3. Compute the exponential Fourier coefficients of $\sin^2(x)$: -$a_n = \frac{1}{\sqrt 2\pi} \int_0^{2\pi} \sin^2(x) e^{-inx} dx$ -and use this to show that -$\int_0^{2\pi} |\sin^2(x)|^2 dx = \sum |a_n|^2.$ +- Try to extend the following functions of a real variable to holomorphic functions defined on the entire complex plane. Is it always possible to do so? What goes wrong? + - $\sinh(z), \cosh(z)$ + - $\frac{z^3}{1 + z^2}$ + - $\sin(z), \cos(z)$ + - $\sqrt{z}$ + - $\log z$ + - $\mathrm{erf}(z)$, the antiderivative of the gaussian $e^{-z^2/2}$ + - $e^{1/z}$ +What is the growth rate of the magnitude of these functions as $z \to \infty$ along the real axis? The imaginary axis? How does the argument change? |