APM346-2018S > Web Bonus Problems

Week 13 -- BP3, 4, 5, 6

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Victor Ivrii:
$\renewcommand{\Re}{\operatorname{Re}}$
There are several connected bonus problems:
a.
Let $(x\pm i0)^{\nu}= \lim _{\varepsilon\to +0} (x\pm \varepsilon i)^{\nu}$. Prove, it exists for $\Re\nu >-1$.
b. Using equality   
\begin{equation}
[(x\pm \varepsilon i)^{\nu}]'= \nu (x\pm \varepsilon i)^{\nu-1},
\label{eq-11.1.12}
\end{equation}
prove that these limits exist (in the sense of distributions) for  $ \nu \ne -1,-2,\ldots$.
c. Using
\begin{equation}
[\ln (x\pm \varepsilon i) ]'=
 (x\pm \varepsilon i)^{-1},
 \label{eq-11.1.13}
\end{equation}
prove that these limits exist (in the sense of distributions) for  $\nu =-1$. Then, using (\ref{eq-11.1.12}), prove that these limits exist (in the sense of distributions) for $\nu=-2,-3,\ldots$.
d. Prove that
\begin{align}
&(x-i0)^{-1} + (x+i0)^{-1}=2 x^{-1},
\label{eq-11.1.14}\\
&(x-i0)^{-1}- (x+i0)^{-1}=2\pi i \delta(x)
\label{eq-11.1.15}
\end{align}
with integral understood in the sense of principal value, use (\ref{eq-11.1.13}) and 
\begin{align}
&\ln (x+i0)+\ln (x-i0)= 2\ln |x|,\\
&\ln (x+i0)-\ln (x-i0)= -2\pi i\theta(-x).
\end{align}
You need to justify these formulae.

Andrew Hardy:
a) Consider
$$ (x\pm i0)^{\nu}= f'$$
Now via definitions $$ (f',\theta) = - (f,\theta') $$ and then f is defined as
 $$ \frac{1}{\nu+1}(x \pm i0)^{\nu+1}  = f $$ This  exists for  $ \Re\nu >-1 $. Therefore the limit exists for this condition.
b) Extending the proof from part a, define $$ (x\pm i0)^{\nu}= f^{(n)}$$ then with the same rules of distribution derivatives I can write $$  [(x\pm \varepsilon i)^{\nu}]^{(n)}= (\nu ! )(x\pm \varepsilon i)^{\nu-n}$$ which is then defined for corresponding $ \Re\nu \neq -n $ where  $ n = 2, 3,\ldots $

Victor Ivrii:
One does not need derivatives to prove that the limit (in $L^1$) $\lim \varepsilon\to +0 (x\pm i\varepsilon)^\nu $ exists and equals
$$
|x|^\nu \left\{\begin{aligned}
&1 &&x>0,\\
&e^{\pm i\pi \nu} && x<0,
\end{aligned}\right.
$$
provided $\Re\nu >-1$ because then this function belongs to $L^1$. The fin starts from (b)

Zhongnan Wu:
I think there is a type of question c). Should be [ln(x+epsilon i)]', here the power of v is 1 and should not be shown.
Also for the second half part of this question, should we use the equality from b) rather than from a)?

Jingxuan Zhang:
Zhongnan,
I think c) is alright as the way it stands. After all \eqref{eq-11.1.12} is deduced from \eqref{eq-11.1.13} so that is also fine...

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