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diff --git a/vol1/1.2.11.3.lyx b/vol1/1.2.11.3.lyx new file mode 100644 index 0000000..637512e --- /dev/null +++ b/vol1/1.2.11.3.lyx @@ -0,0 +1,241 @@ +#LyX 2.4 created this file. For more info see https://www.lyx.org/ +\lyxformat 620 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass book +\use_default_options true +\maintain_unincluded_children no +\language american +\language_package default +\inputencoding utf8 +\fontencoding auto +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_roman_osf false +\font_sans_osf false +\font_typewriter_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\float_placement class +\float_alignment class +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry false +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\use_formatted_ref 0 +\use_minted 0 +\use_lineno 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style english +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tablestyle default +\tracking_changes false +\output_changes false +\change_bars false +\postpone_fragile_content true +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\docbook_table_output 0 +\docbook_mathml_prefix 1 +\end_header + +\begin_body + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TODO 4, + 6 (1 p.) (est. + 0:13:30) +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +rexerc4[HM10] +\end_layout + +\end_inset + +Prove Eq. + (13). +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +answer +\end_layout + +\end_inset + +Since +\begin_inset Formula $u\geq0$ +\end_inset + +, + +\begin_inset Formula $\ln(1+u)\geq0$ +\end_inset + + and +\begin_inset Formula $u=v+\ln(1+u)\geq v$ +\end_inset + +. + On the other hand, + +\begin_inset Formula $x$ +\end_inset + + is fixed with respect to +\begin_inset Formula $v$ +\end_inset + +, + so by monotonicity of the integral, + if +\begin_inset Formula $v\geq1$ +\end_inset + +, +\begin_inset Formula +\begin{multline*} +0\leq\int_{r}^{\infty}x\text{e}^{-xv}\left(1+\frac{1}{u}\right)\text{d}v\leq\int_{r}^{\infty}x\text{e}^{-xv}\left(1+\frac{1}{v}\right)\text{d}v\leq\int_{r}^{\infty}2x\text{e}^{-xv}=\\ +=\left.-2\text{e}^{-xv}\right|_{v=r}^{\infty}=2\text{e}^{-rx}. +\end{multline*} + +\end_inset + +This means +\begin_inset Formula $\int_{r}^{\infty}x\text{e}^{-xv}\left(1+\frac{1}{u}\right)\text{d}v=O(\text{e}^{-rx})$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +rexerc6[HM20] +\end_layout + +\end_inset + +Prove Eq. + (23). +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +answer +\end_layout + +\end_inset + + +\begin_inset Formula +\[ +(n+\alpha)^{n+\beta}=n^{n+\beta}\left(1+\frac{\alpha}{n}\right)^{n+\beta}. +\] + +\end_inset + +For the second factor, +\begin_inset Formula +\begin{multline*} +\ln\left(1+\frac{\alpha}{n}\right)^{n+\beta}=(n+\beta)\ln\left(1+\frac{\alpha}{n}\right)=(n+\beta)\left(\frac{\alpha}{n}-\frac{\alpha^{2}}{2n^{2}}+O(n^{-3})\right)=\\ +=\alpha+\frac{1}{n}\left(\alpha\beta-\frac{\alpha^{2}}{2}\right)+O(n^{-2}), +\end{multline*} + +\end_inset + +so +\begin_inset Formula +\begin{multline*} +n^{n+\beta}\left(1+\frac{\alpha}{n}\right)^{n+\beta}=n^{n+\beta}\text{e}^{\alpha}\exp\left(\frac{1}{n}\left(\alpha\beta-\frac{\alpha^{2}}{2}\right)+O(n^{-2})\right)=\\ +n^{n+\beta}\text{e}^{\alpha}\left(1+\frac{\alpha}{n}\left(\beta-\frac{\alpha}{2}\right)+O(n^{-2})\right). +\end{multline*} + +\end_inset + + +\end_layout + +\end_body +\end_document |