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Diffstat (limited to 'source/know/concept/sokhotski-plemelj-theorem')
| -rw-r--r-- | source/know/concept/sokhotski-plemelj-theorem/index.md | 32 |
1 files changed, 16 insertions, 16 deletions
diff --git a/source/know/concept/sokhotski-plemelj-theorem/index.md b/source/know/concept/sokhotski-plemelj-theorem/index.md index 984a558..66e89bc 100644 --- a/source/know/concept/sokhotski-plemelj-theorem/index.md +++ b/source/know/concept/sokhotski-plemelj-theorem/index.md @@ -9,8 +9,8 @@ categories: layout: "concept" --- -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 $[a, b]$: +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 $$[a, b]$$: $$\begin{aligned} \lim_{\eta \to 0^+} \int_a^b \frac{f(x)}{x + i \eta} \dd{x} @@ -26,7 +26,7 @@ $$\begin{aligned} \end{aligned}$$ To evaluate the real part, -we notice that for $\eta \to 0^+$ the integrand diverges for $x \to 0$, +we notice that for $$\eta \to 0^+$$ the integrand diverges for $$x \to 0$$, and thus split the integral as follows: $$\begin{aligned} @@ -35,7 +35,7 @@ $$\begin{aligned} \end{aligned}$$ This is simply the definition of the -[Cauchy principal value](/know/concept/cauchy-principal-value/) $\mathcal{P}$, +[Cauchy principal value](/know/concept/cauchy-principal-value/) $$\mathcal{P}$$, so the real part is given by: $$\begin{aligned} @@ -45,7 +45,7 @@ $$\begin{aligned} \end{aligned}$$ Meanwhile, in the imaginary part, -we substitute $\eta$ for $1 / m$, and introduce $\pi$: +we substitute $$\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} @@ -54,8 +54,8 @@ $$\begin{aligned} &= \lim_{m \to +\infty} \frac{\pi}{\pi} \int_a^b \frac{m}{1 + m^2 x^2} f(x) \dd{x} \end{aligned}$$ -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 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/): $$\begin{aligned} @@ -76,9 +76,9 @@ $$\begin{aligned} \end{aligned}$$ However, this theorem is often written in the following sloppy way, -where $\eta$ is defined up front to be small, -the integral is hidden, and $f(x)$ is set to $1$. -This awkwardly leaves $\mathcal{P}$ behind: +where $$\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} @@ -87,13 +87,13 @@ $$\begin{aligned} The full, complex version of the Sokhotski-Plemelj theorem evaluates integrals of the following form -over a contour $C$ in the complex plane: +over a contour $$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}$$ -Where $f(z)$ must be [holomorphic](/know/concept/holomorphic-function/). +Where $$f(z)$$ must be [holomorphic](/know/concept/holomorphic-function/). The Sokhotski-Plemelj theorem then states: $$\begin{aligned} @@ -103,8 +103,8 @@ $$\begin{aligned} } \end{aligned}$$ -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, +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 $|\zeta| \to \infty$. +because $$1 / (\zeta \!-\! z) \to 0$$ for $$|\zeta| \to \infty$$. |