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author | Prefetch | 2021-11-01 21:29:02 +0100 |
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committer | Prefetch | 2021-11-01 21:29:02 +0100 |
commit | b090363af28c577bbf9da60d03c82056036588aa (patch) | |
tree | fb3c9dfe1de2e80e33aeb8ff155019c10955db28 | |
parent | f9f062d4382a5f501420ffbe4f19902fe94cf480 (diff) |
Expand knowledge base
-rw-r--r-- | content/know/category/complex-analysis.md | 9 | ||||
-rw-r--r-- | content/know/concept/cauchy-principal-value/index.pdc | 58 | ||||
-rw-r--r-- | content/know/concept/dyson-equation/index.pdc | 174 | ||||
-rw-r--r-- | content/know/concept/holomorphic-function/index.pdc | 1 | ||||
-rw-r--r-- | content/know/concept/kramers-kronig-relations/index.pdc | 1 | ||||
-rw-r--r-- | content/know/concept/martingale/index.pdc | 61 | ||||
-rw-r--r-- | content/know/concept/propagator/index.pdc | 2 | ||||
-rw-r--r-- | content/know/concept/sigma-algebra/index.pdc | 30 | ||||
-rw-r--r-- | content/know/concept/sokhotski-plemelj-theorem/index.pdc | 115 |
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diff --git a/content/know/category/complex-analysis.md b/content/know/category/complex-analysis.md new file mode 100644 index 0000000..fb7cbef --- /dev/null +++ b/content/know/category/complex-analysis.md @@ -0,0 +1,9 @@ +--- +title: "Complex analysis" +firstLetter: "C" +date: 2021-11-01T12:56:57+01:00 +draft: false +layout: "category" +--- + +This page will fill itself. diff --git a/content/know/concept/cauchy-principal-value/index.pdc b/content/know/concept/cauchy-principal-value/index.pdc new file mode 100644 index 0000000..3b59e2b --- /dev/null +++ b/content/know/concept/cauchy-principal-value/index.pdc @@ -0,0 +1,58 @@ +--- +title: "Cauchy principal value" +firstLetter: "C" +publishDate: 2021-11-01 +categories: +- Mathematics + +date: 2021-11-01T12:54:50+01:00 +draft: false +markup: pandoc +--- + +# Cauchy principal value + +The **Cauchy principal value** $\mathcal{P}$, +or just **principal value**, +is a method for integrating problematic functions, +i.e. functions with singularities, +whose integrals would otherwise diverge. + +Consider a function $f(x)$ with a singularity at some finite $x = b$, +which is hampering attempts at integrating it. +To resolve this, we define the Cauchy principal value $\mathcal{P}$ as follows: + +$$\begin{aligned} + \boxed{ + \mathcal{P} \int_a^c f(x) \dd{x} + = \lim_{\varepsilon \to 0^{+}} \!\bigg( \int_a^{b - \varepsilon} f(x) \dd{x} + \int_{b + \varepsilon}^c f(x) \dd{x} \bigg) + } +\end{aligned}$$ + +If $f(x)$ instead has a singularity at postive infinity $+\infty$, +then we define $\mathcal{P}$ as follows: + +$$\begin{aligned} + \boxed{ + \mathcal{P} \int_{a}^\infty f(x) \dd{x} + = \lim_{c \to \infty} \!\bigg( \int_{a}^c f(x) \dd{x} \bigg) + } +\end{aligned}$$ + +And analogously for $-\infty$. +If $f(x)$ has singularities both at $+\infty$ and at $b$, +then we simply combine the two previous cases, +such that $\mathcal{P}$ is given by: + +$$\begin{aligned} + \mathcal{P} \int_{a}^\infty f(x) \:dx + = \lim_{c \to \infty} \lim_{\varepsilon \to 0^{+}} + \!\bigg( \int_{a}^{b - \varepsilon} f(x) \:dx + \int_{b + \varepsilon}^{c} f(x) \:dx \bigg) +\end{aligned}$$ + +And so on, until all problematic singularities have been dealt with. + +In some situations, for example involving +the [Sokhotski-Plemelj theorem](/know/concept/sokhotski-plemelj-theorem/), +the symbol $\mathcal{P}$ is written without an integral, +in which case the calculations are implicitly integrated. diff --git a/content/know/concept/dyson-equation/index.pdc b/content/know/concept/dyson-equation/index.pdc new file mode 100644 index 0000000..7b94124 --- /dev/null +++ b/content/know/concept/dyson-equation/index.pdc @@ -0,0 +1,174 @@ +--- +title: "Dyson equation" +firstLetter: "D" +publishDate: 2021-11-01 +categories: +- Physics +- Quantum mechanics + +date: 2021-11-01T14:57:54+01:00 +draft: false +markup: pandoc +--- + +# Dyson equation + +Consider the time-dependent Schrödinger equation, +describing a wavefunction $\Psi_0(\vb{r}, t)$: + +$$\begin{aligned} + i \hbar \pdv{t} \Psi_0(\vb{r}, t) + = \hat{H}_0(\vb{r}) \: \Psi_0(\vb{r}, t) +\end{aligned}$$ + +By definition, this equation's *fundamental solution* +$G_0(\vb{r}, t; \vb{r}', t')$ satisfies the following: + +$$\begin{aligned} + \Big( i \hbar \pdv{t} - \hat{H}_0(\vb{r}) \Big) G_0(\vb{r}, t; \vb{r}', t') + = \delta(\vb{r} - \vb{r}') \: \delta(t - t') +\end{aligned}$$ + +From this, we define the inverse $\hat{G}{}_0^{-1}(\vb{r}, t)$ +as follows, so that $\hat{G}{}_0^{-1} G_0 = \delta(\vb{r} \!-\! \vb{r}') \: \delta(t \!-\! t')$: + +$$\begin{aligned} + \hat{G}{}_0^{-1}(\vb{r}, t) + &\equiv i \hbar \pdv{t} - \hat{H}_0(\vb{r}) +\end{aligned}$$ + +Note that $\hat{G}{}_0^{-1}$ is an operator, while $G_0$ is a function. +For the sake of consistency, we thus define +the operator $\hat{G}_0(\vb{r}, t)$ +as a multiplication by $G_0$ +and integration over $\vb{r}'$ and $t'$: + +$$\begin{aligned} + \hat{G}_0(\vb{r}, t) \: f + \equiv \iint_{-\infty}^\infty G_0(\vb{r}, t; \vb{r}', t') \: f(\vb{r}', t') \: \dd{\vb{r}}' \dd{t'} +\end{aligned}$$ + +For an arbitrary function $f(\vb{r}, t)$, +so that $\hat{G}{}_0^{-1} \hat{G}_0 = \hat{G}_0 \hat{G}{}_0^{-1} = 1$. +Moving on, the Schrödinger equation can be rewritten like so, +using $\hat{G}{}_0^{-1}$: + +$$\begin{aligned} + \hat{G}{}_0^{-1}(\vb{r}, t) \: \Psi_0(\vb{r}, t) + = 0 +\end{aligned}$$ + +Let us assume that $\hat{H}_0$ is simple, +such that $G_0$ and $\hat{G}{}_0^{-1}$ can be found without issues +by solving the defining equation above. + +Suppose we now perturb this Hamiltonian with +a possibly time-dependent operator $\hat{H}_1(\vb{r}, t)$, +in which case the corresponding fundamental solution +$G(\vb{r}, \vb{r}', t, t')$ satisfies: + +$$\begin{aligned} + \delta(\vb{r} - \vb{r}') \: \delta(t - t') + &= \Big( i \hbar \pdv{t} - \hat{H}_0(\vb{r}) - \hat{H}_1(\vb{r}, t) \Big) G(\vb{r}, t; \vb{r}', t') + \\ + &= \Big( \hat{G}{}_0^{-1}(\vb{r}, t) - \hat{H}_1(\vb{r}, t) \Big) G(\vb{r}, t; \vb{r}', t') +\end{aligned}$$ + +This equation is typically too complicated to solve, +so we would like an easier way to calculate this new $G$. +The perturbed wavefunction $\Psi(\vb{r}, t)$ +satisfies the Schrödinger equation: + +$$\begin{aligned} + \Big( \hat{G}{}_0^{-1}(\vb{r}, t) - \hat{H}_1(\vb{r}, t) \Big) \Psi(\vb{r}, t) + = 0 +\end{aligned}$$ + +We know that $\hat{G}{}_0^{-1} \Psi_0 = 0$, +which we put on the right, +and then we apply $\hat{G}_0$ in front: + +$$\begin{aligned} + \hat{G}_0^{-1} \Psi - \hat{H}_1 \Psi + = \hat{G}_0^{-1} \Psi_0 + \quad \implies \quad + \Psi - \hat{G}_0 \hat{H}_1 \Psi + &= \Psi_0 +\end{aligned}$$ + +This equation is recursive, +so we iteratively insert it into itself. +Note that the resulting equations are the same as those from +[time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/): + +$$\begin{aligned} + \Psi + &= \Psi_0 + \hat{G}_0 \hat{H}_1 \Psi + \\ + &= \Psi_0 + \hat{G}_0 \hat{H}_1 \Psi_0 + \hat{G}_0 \hat{H}_1 \hat{G}_0 \hat{H}_1 \Psi + \\ + &= \Psi_0 + \hat{G}_0 \hat{H}_1 \Psi_0 + \hat{G}_0 \hat{H}_1 \hat{G}_0 \hat{H}_1 \Psi_0 + + \hat{G}_0 \hat{H}_1 \hat{G}_0 \hat{H}_1 \hat{G}_0 \hat{H}_1 \Psi_0 + \: ... + \\ + &= \Psi_0 + \big( \hat{G}_0 + \hat{G}_0 \hat{H}_1 \hat{G}_0 + \hat{G}_0 \hat{H}_1 \hat{G}_0 \hat{H}_1 \hat{G}_0 + \: ... \big) \hat{H}_1 \Psi_0 +\end{aligned}$$ + +The parenthesized expression clearly has the same recursive pattern, +so we denote it by $\hat{G}$ and write the so-called **Dyson equation**: + +$$\begin{aligned} + \boxed{ + \hat{G} + = \hat{G}_0 + \hat{G}_0 \hat{H}_1 \hat{G} + } +\end{aligned}$$ + +Such an iterative scheme is excellent for approximating $\hat{G}(\vb{r}, t)$. +Once a satisfactory accuracy is obtained, +the perturbed wavefunction $\Psi$ can be calculated from: + +$$\begin{aligned} + \boxed{ + \Psi + = \Psi_0 + \hat{G} \hat{H}_1 \Psi_0 + } +\end{aligned}$$ + +This relation is equivalent to the Schrödinger equation. +So now we have the operator $\hat{G}(\vb{r}, t)$, +but what about the fundamental solution function $G(\vb{r}, t; \vb{r}', t')$? +Let us take its definition, multiply it by an arbitrary $f(\vb{r}, t)$, +and integrate over $G$'s second argument pair: + +$$\begin{aligned} + \iint \big( \hat{G}{}_0^{-1} \!-\! \hat{H}_1 \big) G(\vb{r}', t') \: f(\vb{r}', t') \dd{\vb{r}'} \dd{t'} + = \iint^\infty \delta(\vb{r} \!-\! \vb{r}') \: \delta(t \!-\! t') \: f(\vb{r}, t) \dd{\vb{r}'} \dd{t'} + = f +\end{aligned}$$ + +Where we have hidden the arguments $(\vb{r}, t)$ for brevity. +We now apply $\hat{G}_0(\vb{r}, t)$ to this equation +(which contains an integral over $t''$ independent of $t'$): + +$$\begin{aligned} + \hat{G}_0 f + &= \big( \hat{G}_0 \hat{G}{}_0^{-1} - \hat{G}_0 \hat{H}_1 \big) \iint_{-\infty}^\infty G(\vb{r}', t') \: f(\vb{r}', t') \dd{\vb{r}'} \dd{t'} + \\ + &= \big( 1 - \hat{G}_0 \hat{H}_1 \big) \iint_{-\infty}^\infty G(\vb{r}', t') \: f(\vb{r}', t') \dd{\vb{r}'} \dd{t'} +\end{aligned}$$ + +Here, the shape of Dyson's equation is clearly recognizable, +so we conclude that, as expected, the operator $\hat{G}$ +is defined as multiplication by the function $G$ followed by integration: + +$$\begin{aligned} + \hat{G}(\vb{r}, t) \: f(\vb{r}, t) + \equiv \iint_{-\infty}^\infty G(\vb{r}, t; \vb{r}', t') \: f(\vb{r}', t') \: \dd{\vb{r}}' \dd{t'} +\end{aligned}$$ + + + +## References +1. H. Bruus, K. Flensberg, + *Many-body quantum theory in condensed matter physics*, + 2016, Oxford. diff --git a/content/know/concept/holomorphic-function/index.pdc b/content/know/concept/holomorphic-function/index.pdc index 1c2f092..4b7221c 100644 --- a/content/know/concept/holomorphic-function/index.pdc +++ b/content/know/concept/holomorphic-function/index.pdc @@ -4,6 +4,7 @@ firstLetter: "H" publishDate: 2021-02-25 categories: - Mathematics +- Complex analysis date: 2021-02-25T14:40:45+01:00 draft: false diff --git a/content/know/concept/kramers-kronig-relations/index.pdc b/content/know/concept/kramers-kronig-relations/index.pdc index 9b67d60..01e5a3a 100644 --- a/content/know/concept/kramers-kronig-relations/index.pdc +++ b/content/know/concept/kramers-kronig-relations/index.pdc @@ -4,6 +4,7 @@ firstLetter: "K" publishDate: 2021-02-25 categories: - Mathematics +- Complex analysis - Physics - Optics diff --git a/content/know/concept/martingale/index.pdc b/content/know/concept/martingale/index.pdc new file mode 100644 index 0000000..ffc286b --- /dev/null +++ b/content/know/concept/martingale/index.pdc @@ -0,0 +1,61 @@ +--- +title: "Martingale" +firstLetter: "M" +publishDate: 2021-10-31 +categories: +- Mathematics + +date: 2021-10-18T10:01:46+02:00 +draft: false +markup: pandoc +--- + +# Martingale + +A **martingale** is a type of stochastic process +(i.e. a time-indexed [random variable](/know/concept/random-variable/)) +with important and useful properties, +especially for stochastic calculus. + +For a stochastic process $\{ M_t : t \ge 0 \}$ +on a probability space $(\Omega, \mathcal{F}, P)$ with filtration $\{ \mathcal{F}_t \}$ +(see [$\sigma$-algebra](/know/concept/sigma-algebra/)), +then $\{ M_t \}$ is a martingale if it satisfies all of the following: + +1. $M_t$ is $\mathcal{F}_t$-adapted, meaning + the filtration $\mathcal{F}_t$ contains enough information + to reconstruct the current and all past values of $M_t$. +2. For all times $t \ge 0$, the expectation value exists $\mathbf{E}(M_t) < \infty$. +3. For all $s, t$ satisfying $0 \le s \le t$, + the [conditional expectation](/know/concept/conditional-expectation/) + $\mathbf{E}(M_t | \mathcal{F}_s) = M_s$, + meaning the increment $M_t \!-\! M_s$ is always expected + to be zero $\mathbf{E}(M_t \!-\! M_s | \mathcal{F}_s) = 0$. + +The last condition is called the **martingale property**, +and essentially means that a martingale is an unbiased random walk. +Accordingly, the [Wiener process](/know/concept/wiener-process/) $\{ B_t \}$ +(Brownian motion) is a prime example of a martingale +(with respect to its own filtration), +since each of its increments $B_t \!-\! B_s$ has mean $0$ by definition. + +Modifying property (3) leads to two common generalizations. +The stochastic process $\{ M_t \}$ above is a **submartingale** +if the current value is a lower bound for the expectation: + +3. For $0 \le s \le t$, the conditional expectation $\mathbf{E}(M_t | \mathcal{F}_s) \ge M_s$. + +Analogouly, $\{ M_t \}$ is a **supermartingale** +if the current value is an upper bound instead: + +3. For $0 \le s \le t$, the conditional expectation $\mathbf{E}(M_t | \mathcal{F}_s) \le M_s$. + +Clearly, submartingales and supermartingales are *biased* random walks, +since they will tend to increase and decrease with time, respectively. + + + +## References +1. U.F. Thygesen, + *Lecture notes on diffusions and stochastic differential equations*, + 2021, Polyteknisk Kompendie. diff --git a/content/know/concept/propagator/index.pdc b/content/know/concept/propagator/index.pdc index 2f18c4d..517376c 100644 --- a/content/know/concept/propagator/index.pdc +++ b/content/know/concept/propagator/index.pdc @@ -61,7 +61,7 @@ $$\begin{aligned} \end{aligned}$$ Sometimes the name "propagator" is also used to refer to -the so-called *fundamental solution* or *Green's function* $G$ +the so-called *fundamental solution* $G$ of the time-dependent Schrödinger equation, which is related to $K$ by: diff --git a/content/know/concept/sigma-algebra/index.pdc b/content/know/concept/sigma-algebra/index.pdc index 690c4cc..1a459ea 100644 --- a/content/know/concept/sigma-algebra/index.pdc +++ b/content/know/concept/sigma-algebra/index.pdc @@ -43,7 +43,7 @@ is a sub-family of a certain $\mathcal{F}$, which is a valid $\sigma$-algebra in its own right. -## Notable examples +## Notable applications A notable $\sigma$-algebra is the **Borel algebra** $\mathcal{B}(\Omega)$, which is defined when $\Omega$ is a metric space, @@ -54,6 +54,8 @@ and all the subsets of $\mathbb{R}$ obtained by countable sequences of unions and intersections of those intervals. The elements of $\mathcal{B}$ are **Borel sets**. +<hr> + Another example of a $\sigma$-algebra is the **information** obtained by observing a [random variable](/know/concept/random-variable/) $X$. Let $\sigma(X)$ be the information generated by observing $X$, @@ -84,6 +86,32 @@ if $Y$ can always be computed from $X$, i.e. there exists a function $f$ such that $Y(\omega) = f(X(\omega))$ for all $\omega \in \Omega$. +<hr> + +The concept of information can be extended for +stochastic processes (i.e. time-indexed random variables): +if $\{ X_t : t \ge 0 \}$ is a stochastic process, +its **filtration** $\mathcal{F}_t$ contains all +the information generated by $X_t$ up to the current time $t$: + +$$\begin{aligned} + \mathcal{F}_t + = \sigma(X_s : 0 \le s \le t) +\end{aligned}$$ + +In other words, $\mathcal{F}_t$ is the "accumulated" $\sigma$-algebra +of all information extractable from $X_t$, +and hence grows with time: $\mathcal{F}_s \subset \mathcal{F}_t$ for $s < t$. +Given $\mathcal{F}_t$, all values $X_s$ for $s \le t$ can be computed, +i.e. if you know $\mathcal{F}_t$, then the present and past of $X_t$ can be reconstructed. + +Given some filtration $\mathcal{H}_t$, a stochastic process $X_t$ +is said to be *"$\mathcal{H}_t$-adapted"* +if $X_t$'s own filtration $\sigma(X_s : 0 \le s \le t) \subseteq \mathcal{H}_t$, +meaning $\mathcal{H}_t$ contains enough information +to determine the current and past values of $X_t$. +Clearly, $X_t$ is always adapted to its own filtration. + ## References diff --git a/content/know/concept/sokhotski-plemelj-theorem/index.pdc b/content/know/concept/sokhotski-plemelj-theorem/index.pdc new file mode 100644 index 0000000..a482618 --- /dev/null +++ b/content/know/concept/sokhotski-plemelj-theorem/index.pdc @@ -0,0 +1,115 @@ +--- +title: "Sokhotski-Plemelj theorem" +firstLetter: "S" +publishDate: 2021-11-01 +categories: +- Mathematics +- Complex analysis +- Quantum mechanics + +date: 2021-11-01T12:54:37+01:00 +draft: false +markup: pandoc +--- + +# Sokhotski-Plemelj theorem + +The goal is to evaluate integrals of the following form, where $a < 0 < b$, +and $f(x)$ is assumed to be continuous in the integration interval $[a, b]$: + +$$\begin{aligned} + \lim_{\eta \to 0^+} \int_a^b \frac{f(x)}{x + i \eta} \dd{x} +\end{aligned}$$ + +To do so, we start by splitting the integrand +into its real and imaginary parts (limit hidden): + +$$\begin{aligned} + \int_a^b \frac{f(x)}{x + i \eta} \dd{x} + &= \int_a^b \frac{x - i \eta}{x^2 + \eta^2} f(x) \dd{x} + = \int_a^b \bigg( \frac{x}{x^2 + \eta^2} - i \frac{\eta}{x^2 + \eta^2} \bigg) f(x) \dd{x} +\end{aligned}$$ + +To evaluate the real part, +we notice that for $\eta \to 0^+$ the integrand diverges for $x \to 0$, +and thus split the integral as follows: + +$$\begin{aligned} + \lim_{\eta \to 0^+} \int_a^b \frac{x f(x)}{x^2 + \eta^2} \dd{x} + &= \lim_{\eta \to 0^+} \bigg( \int_a^{-\eta} \frac{x f(x)}{x^2 + \eta^2} \dd{x} + \int_\eta^b \frac{x f(x)}{x^2 + \eta^2} \dd{x} \bigg) +\end{aligned}$$ + +This is simply the definition of the +[Cauchy principal value](/know/concept/cauchy-principal-value/) $\mathcal{P}$, +so the real part is given by: + +$$\begin{aligned} + \lim_{\eta \to 0^+} \int_a^b \frac{x f(x)}{x^2 + \eta^2} \dd{x} + &= \mathcal{P} \int_a^b \frac{x f(x)}{x^2} \dd{x} + = \mathcal{P} \int_a^b \frac{f(x)}{x} \dd{x} +\end{aligned}$$ + +Meanwhile, in the imaginary part, +we substitute $\eta$ for $1 / m$, and introduce $\pi$: + +$$\begin{aligned} + \lim_{\eta \to 0^+} \int_a^b \frac{\eta \: f(x)}{x^2 + \eta^2} \dd{x} + &= \lim_{m \to +\infty} \frac{\pi}{\pi} \int_a^b \frac{1/m}{x^2 + 1/m^2} f(x) \dd{x} + \\ + &= \lim_{m \to +\infty} \frac{\pi}{\pi} \int_a^b \frac{m}{1 + m^2 x^2} f(x) \dd{x} +\end{aligned}$$ + +The expression $m / \pi (1 + m^2 x^2)$ is a so-called *nascent delta function*, +meaning that in the limit $m \to +\infty$ it converges to +the [Dirac delta function](/know/concept/dirac-delta-function/): + +$$\begin{aligned} + \lim_{\eta \to 0^+} \int_a^b \frac{\eta \: f(x)}{x^2 + \eta^2} \dd{x} + &= \pi \int_a^b \delta(x) \: f(x) \dd{x} + = \pi f(0) +\end{aligned}$$ + +By combining the real and imaginary parts, +we thus arrive at the (real version of the) +so-called **Sokhotski-Plemelj theorem** of complex analysis: + +$$\begin{aligned} + \boxed{ + \lim_{\eta \to 0^+} \int_a^b \frac{f(x)}{x + i \eta} \dd{x} + = \mathcal{P} \int_a^b \frac{f(x)}{x} \dd{x} - i \pi f(0) + } +\end{aligned}$$ + +However, this theorem is often written in the following sloppy way, +where $\eta$ is defined up front to be small, +the integral is hidden, and $f(x)$ is set to $1$. +This awkwardly leaves $\mathcal{P}$ behind: + +$$\begin{aligned} + \frac{1}{x + i \eta} + = \mathcal{P} \Big( \frac{1}{x} \Big) - i \pi \delta(x) +\end{aligned}$$ + +The full, complex version of the Sokhotski-Plemelj theorem +evaluates integrals of the following form +over a contour $C$ in the complex plane: + +$$\begin{aligned} + \phi(z) = \frac{1}{2 \pi i} \oint_C \frac{f(\zeta)}{\zeta - z} \dd{\zeta} +\end{aligned}$$ + +Where $f(z)$ must be [holomorphic](/know/concept/holomorphic-function/). +The Sokhotski-Plemelj theorem then states: + +$$\begin{aligned} + \boxed{ + \lim_{w \to z} \phi(w) + = \frac{1}{2 \pi i} \mathcal{P} \oint_C \frac{f(\zeta)}{\zeta - z} \dd{\zeta} \pm \frac{f(z)}{2} + } +\end{aligned}$$ + +Where the sign is positive if $z$ is inside $C$, and negative if it is outside. +The real version follows by letting $C$ follow the whole real axis, +making $C$ an infinitely large semicircle, +so that the integrand vanishes away from the real axis, +because $1 / (\zeta \!-\! z) \to 0$ for $|\zeta| \to \infty$. |