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diff --git a/content/know/concept/conditional-expectation/index.pdc b/content/know/concept/conditional-expectation/index.pdc index 5bcc152..5a8f07e 100644 --- a/content/know/concept/conditional-expectation/index.pdc +++ b/content/know/concept/conditional-expectation/index.pdc @@ -77,10 +77,10 @@ $$\begin{aligned} Recall that because $Y$ is a random variable, $\mathbf{E}[X|Y] = f(Y)$ is too. In other words, $f$ maps $Y$ to another random variable, -which, due to the *Doob-Dynkin lemma* -(see [$\sigma$-algebra](/know/concept/sigma-algebra/)), -must mean that $\mathbf{E}[X|Y]$ is measurable with respect to $\sigma(Y)$. -Intuitively, this makes some sense: +which, thanks to the *Doob-Dynkin lemma* +(see [random variable](/know/concept/random-variable/)), +means that $\mathbf{E}[X|Y]$ is measurable with respect to $\sigma(Y)$. +Intuitively, this makes sense: $\mathbf{E}[X|Y]$ cannot contain more information about events than the $Y$ it was calculated from. diff --git a/content/know/concept/holomorphic-function/index.pdc b/content/know/concept/holomorphic-function/index.pdc index 4b7221c..3e3984a 100644 --- a/content/know/concept/holomorphic-function/index.pdc +++ b/content/know/concept/holomorphic-function/index.pdc @@ -193,60 +193,3 @@ this proof works inductively for all higher orders $n$. </div> </div> - -## Residue theorem - -A function $f(z)$ is **meromorphic** if it is holomorphic except in -a finite number of **simple poles**, which are points $z_p$ where -$f(z_p)$ diverges, but where the product $(z - z_p) f(z)$ is non-zero and -still holomorphic close to $z_p$. - -The **residue** $R_p$ of a simple pole $z_p$ is defined as follows, and -represents the rate at which $f(z)$ diverges close to $z_p$: - -$$\begin{aligned} - \boxed{ - R_p = \lim_{z \to z_p} (z - z_p) f(z) - } -\end{aligned}$$ - -**Cauchy's residue theorem** generalizes Cauchy's integral theorem -to meromorphic functions, and states that the integral of a contour $C$ -depends on the simple poles $p$ it encloses: - -$$\begin{aligned} - \boxed{ - \oint_C f(z) \dd{z} = i 2 \pi \sum_{p} R_p - } -\end{aligned}$$ - -<div class="accordion"> -<input type="checkbox" id="proof-res-theorem"/> -<label for="proof-res-theorem">Proof</label> -<div class="hidden"> -<label for="proof-res-theorem">Proof.</label> -From the definition of a meromorphic function, -we know that we can decompose $f(z)$ like so, -where $h(z)$ is holomorphic and $p$ are all its poles: - -$$\begin{aligned} - f(z) = h(z) + \sum_{p} \frac{R_p}{z - z_p} -\end{aligned}$$ - -We integrate this over a contour $C$ which contains all poles, and apply -both Cauchy's integral theorem and Cauchy's integral formula to get: - -$$\begin{aligned} - \oint_C f(z) \dd{z} - &= \oint_C h(z) \dd{z} + \sum_{p} R_p \oint_C \frac{1}{z - z_p} \dd{z} - = \sum_{p} R_p \: 2 \pi i -\end{aligned}$$ -</div> -</div> - -This theorem might not seem very useful, -but in fact, thanks to some clever mathematical magic, -it allows us to evaluate many integrals along the real axis, -most notably [Fourier transforms](/know/concept/fourier-transform/). -It can also be used to derive the [Kramers-Kronig relations](/know/concept/kramers-kronig-relations). - diff --git a/content/know/concept/ito-calculus/index.pdc b/content/know/concept/ito-calculus/index.pdc index 3527b1d..7a80e2f 100644 --- a/content/know/concept/ito-calculus/index.pdc +++ b/content/know/concept/ito-calculus/index.pdc @@ -60,6 +60,9 @@ $$\begin{aligned} An Itō process $X_t$ is said to satisfy this equation if $f(X_t, t) = F_t$ and $g(X_t, t) = G_t$, in which case $X_t$ is also called an **Itō diffusion**. +All Itō diffusions are [Markov processes](/know/concept/markov-process/), +since only the current value of $X_t$ determines the future, +and $B_t$ is also a Markov process. ## Itō's lemma @@ -80,7 +83,7 @@ known as **Itō's lemma**: $$\begin{aligned} \boxed{ \dd{Y_t} - = \pdv{h}{t} \dd{t} + \bigg( \pdv{h}{x} F_t + \frac{1}{2} G_t^2 \pdv[2]{h}{x} \bigg) \dd{t} + \pdv{h}{x} G_t \dd{B_t} + = \bigg( \pdv{h}{t} + \pdv{h}{x} F_t + \frac{1}{2} \pdv[2]{h}{x} G_t^2 \bigg) \dd{t} + \pdv{h}{x} G_t \dd{B_t} } \end{aligned}$$ diff --git a/content/know/concept/kolmogorov-equations/index.pdc b/content/know/concept/kolmogorov-equations/index.pdc new file mode 100644 index 0000000..331d803 --- /dev/null +++ b/content/know/concept/kolmogorov-equations/index.pdc @@ -0,0 +1,209 @@ +--- +title: "Kolmogorov equations" +firstLetter: "K" +publishDate: 2021-11-14 +categories: +- Mathematics +- Statistics + +date: 2021-11-13T21:05:30+01:00 +draft: false +markup: pandoc +--- + +# Kolmogorov equations + +Consider the following general [Itō diffusion](/know/concept/ito-calculus/) +$X_t \in \mathbb{R}$, which is assumed to satisfy +the conditions for unique existence on the entire time axis: + +$$\begin{aligned} + \dd{X}_t + = f(X_t, t) \dd{t} + g(X_t, t) \dd{B_t} +\end{aligned}$$ + +Let $\mathcal{F}_t$ be the filtration to which $X_t$ is adapted, +then we define $Y_s$ as shown below, +namely as the [conditional expectation](/know/concept/conditional-expectation/) +of $h(X_t)$, for an arbitrary bounded function $h(x)$, +given the information $\mathcal{F}_s$ available at time $s \le t$. +Because $X_t$ is a [Markov process](/know/concept/markov-process/), +$Y_s$ must be $X_s$-measurable, +so it is a function $k$ of $X_s$ and $s$: + +$$\begin{aligned} + Y_s + \equiv \mathbf{E}[h(X_t) | \mathcal{F}_s] + = \mathbf{E}[h(X_t) | X_s] + = k(X_s, s) +\end{aligned}$$ + +Consequently, we can apply Itō's lemma to find $\dd{Y_s}$ +in terms of $k$, $f$ and $g$: + +$$\begin{aligned} + \dd{Y_s} + &= \bigg( \pdv{k}{s} + \pdv{k}{x} f + \frac{1}{2} \pdv[2]{k}{x} g^2 \bigg) \dd{s} + \pdv{k}{x} g \dd{B_s} + \\ + &= \bigg( \pdv{k}{s} + \hat{L} k \bigg) \dd{s} + \pdv{k}{x} g \dd{B_s} +\end{aligned}$$ + +Where we have defined the linear operator $\hat{L}$ +to have the following action on $k$: + +$$\begin{aligned} + \hat{L} k + \equiv \pdv{k}{x} f + \frac{1}{2} \pdv[2]{k}{x} g^2 +\end{aligned}$$ + +At this point, we need to realize that $Y_s$ is +a [martingale](/know/concept/martingale/) with respect to $\mathcal{F}_s$, +since $Y_s$ is $\mathcal{F}_s$-adapted and finite, +and it satisfies the martingale property, +for $r \le s \le t$: + +$$\begin{aligned} + \mathbf{E}[Y_s | \mathcal{F}_r] + = \mathbf{E}\Big[ \mathbf{E}[h(X_t) | \mathcal{F}_s] \Big| \mathcal{F}_r \Big] + = \mathbf{E}\big[ h(X_t) \big| \mathcal{F}_r \big] + = Y_r +\end{aligned}$$ + +Where we used the tower property of conditional expectations, +because $\mathcal{F}_r \subset \mathcal{F}_s$. +However, an Itō diffusion can only be a martingale +if its drift term (the one containing $\dd{s}$) vanishes, +so, looking at $\dd{Y_s}$, we must demand that: + +$$\begin{aligned} + \pdv{k}{s} + \hat{L} k + = 0 +\end{aligned}$$ + +Because $k(X_s, s)$ is a Markov process, +we can write it with a transition density $p(s, X_s; t, X_t)$, +where in this case $s$ and $X_s$ are given initial conditions, +$t$ is a parameter, and the terminal state $X_t$ is a random variable. +We thus have: + +$$\begin{aligned} + k(x, s) + = \int_{-\infty}^\infty p(s, x; t, y) \: h(y) \dd{y} +\end{aligned}$$ + +We insert this into the equation that we just derived for $k$, yielding: + +$$\begin{aligned} + 0 + = \int_{-\infty}^\infty \!\! \Big( \pdv{s} p(s, x; t, y) + \hat{L} p(s, x; t, y) \Big) h(y) \dd{y} +\end{aligned}$$ + +Because $h$ is arbitrary, and this must be satisfied for all $h$, +the transition density $p$ fulfills: + +$$\begin{aligned} + 0 + = \pdv{s} p(s, x; t, y) + \hat{L} p(s, x; t, y) +\end{aligned}$$ + +Here, $t$ is a known parameter and $y$ is a "known" integration variable, +leaving only $s$ and $x$ as free variables for us to choose. +We therefore define the **likelihood function** $\psi(s, x)$, +which gives the likelihood of an initial condition $(s, x)$ +given that the terminal condition is $(t, y)$: + +$$\begin{aligned} + \boxed{ + \psi(s, x) + \equiv p(s, x; t, y) + } +\end{aligned}$$ + +And from the above derivation, +we conclude that $\psi$ satisfies the following PDE, +known as the **backward Kolmogorov equation**: + +$$\begin{aligned} + \boxed{ + - \pdv{\psi}{s} + = \hat{L} \psi + = f \pdv{\psi}{x} + \frac{1}{2} g^2 \pdv[2]{\psi}{x} + } +\end{aligned}$$ + +Moving on, we can define the traditional +**probability density function** $\phi(t, y)$ from the transition density $p$, +by fixing the initial $(s, x)$ +and leaving the terminal $(t, y)$ free: + +$$\begin{aligned} + \boxed{ + \phi(t, y) + \equiv p(s, x; t, y) + } +\end{aligned}$$ + +With this in mind, for $(s, x) = (0, X_0)$, +the unconditional expectation $\mathbf{E}[Y_t]$ +(i.e. the conditional expectation without information) +will be constant in time, because $Y_t$ is a martingale: + +$$\begin{aligned} + \mathbf{E}[Y_t] + = \mathbf{E}[k(X_t, t)] + = \int_{-\infty}^\infty k(y, t) \: \phi(t, y) \dd{y} + = \braket{k}{\phi} + = \mathrm{const} +\end{aligned}$$ + +This integral has the form of an inner product, +so we switch to [Dirac notation](/know/concept/dirac-notation/). +We differentiate with respect to $t$, +and use the backward equation $\pdv*{k}{t} + \hat{L} k = 0$: + +$$\begin{aligned} + 0 + = \pdv{t} \braket{k}{\phi} + = \braket{k}{\pdv{\phi}{t}} + \braket{\pdv{k}{t}}{\phi} + = \braket{k}{\pdv{\phi}{t}} - \braket{\hat{L} k}{\phi} + = \braket{k}{\pdv{\phi}{t} - \hat{L}{}^\dagger \phi} +\end{aligned}$$ + +Where $\hat{L}{}^\dagger$ is by definition the adjoint operator of $\hat{L}$, +which we calculate using partial integration, +where all boundary terms vanish thanks to the *existence* of $X_t$; +in other words, $X_t$ cannot reach infinity at any finite $t$, +so the integrand must decay to zero for $|y| \to \infty$: + +$$\begin{aligned} + \braket{\hat{L} k}{\phi} + &= \int_{-\infty}^\infty \pdv{k}{y} f \phi + \frac{1}{2} \pdv[2]{k}{y} g^2 \phi \dd{y} + \\ + &= \bigg[ k f \phi + \frac{1}{2} \pdv{k}{y} g^2 \phi \bigg]_{-\infty}^\infty + - \int_{-\infty}^\infty k \pdv{y}(f \phi) + \frac{1}{2} \pdv{k}{y} \pdv{y}(g^2 \phi) \dd{y} + \\ + &= \bigg[ -\frac{1}{2} k g^2 \phi \bigg]_{-\infty}^\infty + + \int_{-\infty}^\infty - k \pdv{y}(f \phi) + \frac{1}{2} k \pdv[2]{y}(g^2 \phi) \dd{y} + \\ + &= \int_{-\infty}^\infty k \: \big( \hat{L}{}^\dagger \phi \big) \dd{y} + = \braket{k}{\hat{L}{}^\dagger \phi} +\end{aligned}$$ + +Since $k$ is arbitrary, and $\pdv*{\braket{k}{\phi}}{t} = 0$ for all $k$, +we thus arrive at the **forward Kolmogorov equation**, +describing the evolution of the probability density $\phi(t, y)$: + +$$\begin{aligned} + \boxed{ + \pdv{\phi}{t} + = \hat{L}{}^\dagger \phi + = - \pdv{y}(f \phi) + \frac{1}{2} \pdv[2]{y}(g^2 \phi) + } +\end{aligned}$$ + + + +## References +1. U.H. Thygesen, + *Lecture notes on diffusions and stochastic differential equations*, + 2021, Polyteknisk Kompendie. diff --git a/content/know/concept/markov-process/index.pdc b/content/know/concept/markov-process/index.pdc new file mode 100644 index 0000000..536aa00 --- /dev/null +++ b/content/know/concept/markov-process/index.pdc @@ -0,0 +1,66 @@ +--- +title: "Markov process" +firstLetter: "M" +publishDate: 2021-11-14 +categories: +- Mathematics + +date: 2021-11-13T21:05:21+01:00 +draft: false +markup: pandoc +--- + +# Markov process + +Given a [stochastic process](/know/concept/stochastic-process/) +$\{X_t : t \ge 0\}$ on a filtered probability space +$(\Omega, \mathcal{F}, \{\mathcal{F}_t\}, P)$, +it is said to be a **Markov process** +if it satisfies the following requirements: + +1. $X_t$ is $\mathcal{F}_t$-adapted, + meaning that the current and all past values of $X_t$ + can be reconstructed from the filtration $\mathcal{F}_t$. +2. For some function $h(x)$, + the [conditional expectation](/know/concept/conditional-expectation/) + $\mathbf{E}[h(X_t) | \mathcal{F}_s] = \mathbf{E}[h(X_t) | X_s]$, + i.e. at time $s \le t$, the expectation of $h(X_t)$ depends only on the current $X_s$. + Note that $h$ must be bounded and *Borel-measurable*, + meaning $\sigma(h(X_t)) \subseteq \mathcal{F}_t$. + +This last condition is called the **Markov property**, +and demands that the future of $X_t$ does not depend on the past, +but only on the present $X_s$. + +If both $t$ and $X_t$ are taken to be discrete, +then $X_t$ is known as a **Markov chain**. +This brings us to the concept of the **transition probability** +$P(X_t \in A | X_s = x)$, which describes the probability that +$X_t$ will be in a given set $A$, if we know that currently $X_s = x$. + +If $t$ and $X_t$ are continuous, we can often (but not always) express $P$ +using a **transition density** $p(s, x; t, y)$, +which gives the probability density that the initial condition $X_s = x$ +will evolve into the terminal condition $X_t = y$. +Then the transition probability $P$ can be calculated like so, +where $B$ is a given Borel set (see [$\sigma$-algebra](/know/concept/sigma-algebra/)): + +$$\begin{aligned} + P(X_t \in B | X_s = x) + = \int_B p(s, x; t, y) \dd{y} +\end{aligned}$$ + +A prime examples of a continuous Markov process is +the [Wiener process](/know/concept/wiener-process/). +Note that this is also a [martingale](/know/concept/martingale/): +often, a Markov process happens to be a martingale, or vice versa. +However, those concepts are not to be confused: +the Markov property does not specify *what* the expected future must be, +and the martingale property says nothing about the history-dependence. + + + +## References +1. U.H. Thygesen, + *Lecture notes on diffusions and stochastic differential equations*, + 2021, Polyteknisk Kompendie. diff --git a/content/know/concept/martingale/index.pdc b/content/know/concept/martingale/index.pdc index 21fa918..41c2709 100644 --- a/content/know/concept/martingale/index.pdc +++ b/content/know/concept/martingale/index.pdc @@ -37,6 +37,14 @@ Accordingly, the [Wiener process](/know/concept/wiener-process/) $B_t$ (Brownian motion) is an example of a martingale, since each of its increments $B_t \!-\! B_s$ has mean $0$ by definition. +Martingales are easily confused with +[Markov processes](/know/concept/markov-process/), +because stochastic processes will often be both, +e.g. the Wiener process. +However, these are distinct concepts: +the martingale property says nothing about history-dependence, +and the Markov property does not say *what* the future expectation should be. + 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: diff --git a/content/know/concept/matsubara-sum/index.pdc b/content/know/concept/matsubara-sum/index.pdc new file mode 100644 index 0000000..91183e6 --- /dev/null +++ b/content/know/concept/matsubara-sum/index.pdc @@ -0,0 +1,148 @@ +--- +title: "Matsubara sum" +firstLetter: "M" +publishDate: 2021-11-13 +categories: +- Physics +- Quantum mechanics + +date: 2021-11-05T15:19:38+01:00 +draft: false +markup: pandoc +--- + +# Matsubara sum + +A **Matsubara sum** is a summation of the following form, +which notably appears as the inverse +[Fourier transform](/know/concept/fourier-transform/) of the +[Matsubara Green's function](/know/concept/matsubara-greens-function/): + +$$\begin{aligned} + S_{B,F} + = \frac{1}{\hbar \beta} \sum_{i \omega_n} g(i \omega_n) \: e^{i \omega_n \tau} +\end{aligned}$$ + +Where $i \omega_n$ are the Matsubara frequencies +for bosons ($B$) or fermions ($F$), +and $g(z)$ is a function on the complex plane +that is [holomorphic](/know/concept/holomorphic-function/) +except for a known set of simple poles, +and $\tau$ is a real parameter +(e.g. the [imaginary time](/know/concept/imaginary-time/)) +satisfying $-\hbar \beta < \tau < \hbar \beta$. + +Now, consider the following integral +over a (for now) unspecified counter-clockwise contour $C$, +with a (for now) unspecified weighting function $h(z)$: + +$$\begin{aligned} + \oint_C \frac{g(z) h(z)}{2 \pi i} e^{z \tau} \dd{z} + = \sum_{z_p} e^{z_p \tau} \: \underset{z \to z_p}{\mathrm{Res}}\big( g(z) h(z) \big) +\end{aligned}$$ + +Where we have applied the residue theorem +to get a sum over all simple poles $z_p$ +of either $g$ or $h$ (but not both) enclosed by $C$. +Clearly, we could make this look like a Matsubara sum, +if we choose $h$ such that it has poles at $i \omega_n$. + +Therefore, we choose the weighting function $h(z)$ as follows, +where $n_B(z)$ is the [Bose-Einstein distribution](/know/concept/bose-einstein-distribution/), +and $n_F(z)$ is the [Fermi-Dirac distribution](/know/concept/fermi-dirac-distribution/): + +$$\begin{aligned} + h(z) + = + \begin{cases} + n_{B,F}(z) & \mathrm{if}\; \tau \ge 0 + \\ + -n_{B,F}(-z) & \mathrm{if}\; \tau \le 0 + \end{cases} + \qquad \qquad + n_{B,F}(z) + = \frac{1}{e^{\hbar \beta z} \mp 1} +\end{aligned}$$ + +The distinction between the signs of $\tau$ is needed +to ensure that the integrand $h(z) e^{z \tau}$ decays for $|z| \to \infty$, +both for $\Re(z) > 0$ and $\Re(z) < 0$. +This choice of $h$ indeed has poles at the respective +Matsubara frequencies $i \omega_n$ of bosons and fermions, +and the residues are: + +$$\begin{aligned} + \underset{z \to i \omega_n}{\mathrm{Res}}\!\big( n_B(z) \big) + &= \lim_{z \to i \omega_n}\!\bigg( \frac{z - i \omega_n}{e^{\hbar \beta z} - 1} \bigg) + = \lim_{\eta \to 0}\!\bigg( \frac{i \omega_n + \eta - i \omega_n}{e^{i \hbar \beta \omega_n} e^{\hbar \beta \eta} - 1} \bigg) + \\ + &= \lim_{\eta \to 0}\!\bigg( \frac{\eta}{e^{\hbar \beta \eta} - 1} \bigg) + = \lim_{\eta \to 0}\!\bigg( \frac{\eta}{1 + \hbar \beta \eta - 1} \bigg) + = \frac{1}{\hbar \beta} + \\ + \underset{z \to i \omega_n}{\mathrm{Res}}\!\big( n_F(z) \big) + &= \lim_{z \to i \omega_n}\!\bigg( \frac{z - i \omega_n}{e^{\hbar \beta z} + 1} \bigg) + = \lim_{\eta \to 0}\!\bigg( \frac{i \omega_n + \eta - i \omega_n}{e^{i \hbar \beta \omega_n} e^{\hbar \beta \eta} + 1} \bigg) + \\ + &= \lim_{\eta \to 0}\!\bigg( \frac{\eta}{e^{\hbar \beta \eta} + 1} \bigg) + = \lim_{\eta \to 0}\!\bigg( \frac{\eta}{- 1 - \hbar \beta \eta + 1} \bigg) + = - \frac{1}{\hbar \beta} +\end{aligned}$$ + +In the definition of $h$, the sign flip for $\tau \le 0$ +is introduced because negating the argument also negates the residues, +i.e. $\mathrm{Res}\big( n_F(-z) \big) = -\mathrm{Res}\big( n_F(z) \big)$. +With this $h$, our contour integral can be rewritten as follows: + +$$\begin{aligned} + \oint_C \frac{g(z) h(z)}{2 \pi i} e^{z \tau} \dd{z} + &= \sum_{z_p} e^{z_p \tau} n_{B,F}(z_p) \: \underset{z \to z_p}{\mathrm{Res}}\big( g(z) \big) + + \sum_{i \omega_n} e^{i \omega_n \tau} g(i \omega_n) \: \underset{z \to i \omega_n}{\mathrm{Res}}\!\big( n_{B,F}(z) \big) + \\ + &= \sum_{z_p} e^{z_p \tau} n_{B,F}(z_p) \: \underset{z \to z_p}{\mathrm{Res}}\big( g(z) \big) + \pm \frac{1}{\hbar \beta} \sum_{i \omega_n} g(i \omega_n) \: e^{i \omega_n \tau} +\end{aligned}$$ + +Where $+$ is for bosons, and $-$ for fermions. +Here, we recognize the last term as the Matsubara sum $S_{F,B}$, +for which we isolate, yielding: + +$$\begin{aligned} + S_{B,F} + = \mp \sum_{z_p} e^{z_p \tau} n_{B,F}(z_p) \: \underset{z \to z_p}{\mathrm{Res}}\big( g(z) \big) + \pm \oint_C \frac{g(z) h(z)}{2 \pi i} e^{z \tau} \dd{z} +\end{aligned}$$ + +Now we must choose $C$. Assuming $g(z)$ does not interfere, +we know that $h(z) e^{z \tau}$ decays to zero +for $|z| \to \infty$, so a useful choice would be a circle of radius $R$. +If we then let $R \to \infty$, the contour encloses +the whole complex plane, including all of the integrand's poles. +However, thanks to the integrand's decay, +the resulting contour integral must vanish: + +$$\begin{aligned} + C + = R e^{i \theta} + \quad \implies \quad + \lim_{R \to \infty} + \oint_C g(z) \: h(z) \: e^{z \tau} \dd{z} + = 0 +\end{aligned}$$ + +We thus arrive at the following results +for bosonic and fermionic Matsubara sums $S_{B,F}$: + +$$\begin{aligned} + \boxed{ + S_{B,F} + = \mp \sum_{z_p} e^{z_p \tau} n_{B,F}(z_p) \: \underset{{z \to z_p}}{\mathrm{Res}}\big(g(z)\big) + } +\end{aligned}$$ + + + +## References +1. H. Bruus, K. Flensberg, + *Many-body quantum theory in condensed matter physics*, + 2016, Oxford. diff --git a/content/know/concept/residue-theorem/index.pdc b/content/know/concept/residue-theorem/index.pdc new file mode 100644 index 0000000..02a8ece --- /dev/null +++ b/content/know/concept/residue-theorem/index.pdc @@ -0,0 +1,77 @@ +--- +title: "Residue theorem" +firstLetter: "R" +publishDate: 2021-11-13 +categories: +- Mathematics +- Complex analysis + +date: 2021-11-13T20:51:13+01:00 +draft: false +markup: pandoc +--- + +# Residue theorem + +A function $f(z)$ is **meromorphic** if it is +[holomorphic](/know/concept/holomorphic-function/) +except in a finite number of **simple poles**, +which are points $z_p$ where $f(z_p)$ diverges, +but where the product $(z - z_p) f(z)$ is non-zero +and still holomorphic close to $z_p$. +In other words, $f(z)$ can be approximated close to $z_p$: + +$$\begin{aligned} + f(z) + \approx \frac{R_p}{z - z_p} +\end{aligned}$$ + +Where the **residue** $R_p$ of a simple pole $z_p$ is defined as follows, and +represents the rate at which $f(z)$ diverges close to $z_p$: + +$$\begin{aligned} + \boxed{ + R_p = \lim_{z \to z_p} (z - z_p) f(z) + } +\end{aligned}$$ + +**Cauchy's residue theorem** for meromorphic functions +is a generalization of Cauchy's integral theorem for holomorphic functions, +and states that the integral on a contour $C$ +purely depends on the simple poles $z_p$ enclosed by $C$: + +$$\begin{aligned} + \boxed{ + \oint_C f(z) \dd{z} = i 2 \pi \sum_{z_p} R_p + } +\end{aligned}$$ + +<div class="accordion"> +<input type="checkbox" id="proof-res-theorem"/> +<label for="proof-res-theorem">Proof</label> +<div class="hidden"> +<label for="proof-res-theorem">Proof.</label> +From the definition of a meromorphic function, +we know that we can decompose $f(z)$ like so, +where $h(z)$ is holomorphic and $z_p$ are all its poles: + +$$\begin{aligned} + f(z) = h(z) + \sum_{z_p} \frac{R_p}{z - z_p} +\end{aligned}$$ + +We integrate this over a contour $C$ which contains all poles, and apply +both Cauchy's integral theorem and Cauchy's integral formula to get: + +$$\begin{aligned} + \oint_C f(z) \dd{z} + &= \oint_C h(z) \dd{z} + \sum_{p} R_p \oint_C \frac{1}{z - z_p} \dd{z} + = \sum_{p} R_p \: 2 \pi i +\end{aligned}$$ +</div> +</div> + +This theorem might not seem very useful, +but in fact, by cleverly choosing the contour $C$, +it lets us evaluate many integrals along the real axis, +most notably [Fourier transforms](/know/concept/fourier-transform/). +It can also be used to derive the [Kramers-Kronig relations](/know/concept/kramers-kronig-relations). diff --git a/content/know/concept/wiener-process/index.pdc b/content/know/concept/wiener-process/index.pdc index f8610a2..dc3892d 100644 --- a/content/know/concept/wiener-process/index.pdc +++ b/content/know/concept/wiener-process/index.pdc @@ -60,6 +60,12 @@ $$\begin{aligned} = \infty \end{aligned}$$ +Furthermore, the Wiener process is a good example +of both a [martingale](/know/concept/martingale/) +and a [Markov process](/know/concept/markov-process/), +since each increment has mean zero (so it is a martingale), +and all increments are independent (so it is a Markov process). + ## References |