预备：Lp、卷积与插值

\usepage{tikz}

\begin{center}
    \begin{tikzpicture}[>={Stealth[length=0.3cm]},
        decoration={markings,
        mark= at position .1 with {\arrow{<}},
        mark= at position .45 with {\arrow{<}},
        mark= at position .83 with {\arrow{<}},
        }
    ]
        \def\angle{40}
        \def\bigradius{5}
        \def\littleradius{1}
        \filldraw[postaction = {decorate}, thick ,fill=gray!40]
            (\angle:\littleradius) node[right]{<span data-mx-katex="%5Cepsilon"></span>}
            -- (\angle:\bigradius) node[right]{<span data-mx-katex="R"></span>}
            arc(\angle:0:\bigradius) node[below]{<span data-mx-katex="R"></span>}
            -- (\littleradius,0) node[below]{<span data-mx-katex="%5Cepsilon"></span>}
            arc (0:\angle:\littleradius) -- cycle;

        \draw[dashed,thick] (0,0) node[xshift=15pt, yshift=6pt]{<span data-mx-katex="%5Ctheta"></span>} -- (\angle:\littleradius);
        \node at(3,-0.3){<span data-mx-katex="%5CGamma_1"></span>};
        \node at(5,2){<span data-mx-katex="%5CGamma_2"></span>};
        \node at(2.5,2.5){<span data-mx-katex="%5CGamma_3"></span>};
        \node at(1.5,0.5){<span data-mx-katex="%5CGamma_4"></span>};
        \draw[-Latex] (-0.1*\bigradius,0) -- (1.2*\bigradius,0) node[below]{<span data-mx-katex="x"></span>} ;
        \draw[-Latex] (0,-0.1*\bigradius) -- (0,4) node[right]{<span data-mx-katex="iy"></span>};
    \end{tikzpicture}
\end{center}

\begin{center}
    \begin{tikzpicture}
        \def\gap{0.2}
        \def\bigradius{3}
        \def\littleradius{0.5}
        % Axes

        \draw[->](-1,-3) -- (5,3) -- (3,3) -- (-3,-3) -- (-1,-3) --cycle;
        \fill[gray!40](-1,-3) -- (5,3) -- (3,3) -- (-3,-3) -- (-1,-3) --cycle;


        \draw [thick,->] (-4, 0) -- (6,0);
        \draw [thick,->] (0, -4) -- (0,4);

        \node at (0.5,4){<span data-mx-katex="iy"></span>};
        \node at (6.2,0.3){<span data-mx-katex="x"></span>};
        \node at (1.3,-1.3){<span data-mx-katex="%5CGamma_1"></span>};
        \node at (4,3.3){<span data-mx-katex="%5CGamma_2"></span>};
        \node at (1.1,1.6){<span data-mx-katex="%5CGamma_3"></span>};
        \node at (-2,-3.3){<span data-mx-katex="%5CGamma_4"></span>};
        \draw(1,0) node[shape=circle,draw,inner sep=1pt]{};
        \node at (1.2,0.3){<span data-mx-katex="%5Cfrac%7B1%7D%7B2%7D"></span>};
        \draw(3,0) node[shape=circle,draw,inner sep=1pt]{};
        \node at (3.2,-0.3){<span data-mx-katex="R"></span>};
        \draw(5,0) node[shape=circle,draw,inner sep=1pt]{};
        \node at (5.2,-0.3){<span data-mx-katex="R%2B1"></span>};
        \draw(0,3) node[shape=circle,draw,inner sep=1pt]{};
        \node at (0.3,3.3){<span data-mx-katex="iR"></span>};
        \draw[dashed, thick] (0,3)--(3,3)--(3,0);
        \draw[dashed, thick] (5,0)--(5,3);
        %\draw[fill=gray,domain=-3:3]plot({\x-2},\x);

    \end{tikzpicture}
\end{center}

\begin{center}
    \begin{tikzpicture}[>={Stealth[length=0.3cm]},
        decoration={markings,
        mark= at position .15 with {\arrow{>}},
        mark= at position .3 with {\arrow{>}},
        %mark= at position .42 with {\arrow{>}},
        mark= at position .49 with {\arrow{>}},
        mark= at position .63 with {\arrow{>}},
        mark= at position 0.9 with {\arrow{>}},
        }]
        \def\radius{3.4}
        \def\bigradius{3.4}
        \def\littleradius{0.5}
        \def\angle{60}
        \def\gap{0.3}
    \draw[dashed,thick]
        let
            \n0 = {cos(\angle)},
        in
            (\radius*\n0,-\radius*1.2) node[right]{<span data-mx-katex="c%20-%20i%5Cinfty"></span>}-- (\radius*\n0,\radius*1.2)node[right]{<span data-mx-katex="c%2Bi%5Cinfty"></span>};
    % contour
    \filldraw[postaction = {decorate}, thick ,fill=gray!40]
    let
    \n1 = {asin(\gap/2/\bigradius)},
    \n2 = {asin(\gap/2/\littleradius)}
    in
    (0.5*\bigradius,\bigradius)node[right]{<span data-mx-katex="B"></span>}--(0,\bigradius) arc(90:180-\n1:\bigradius) --(-\n2:-\littleradius) arc (-\n2:-(360-\n2):-\littleradius) --(\n1:-\bigradius) arc(180+\n1:270:\bigradius)--(0.5*\bigradius,-\bigradius)node[right]{<span data-mx-katex="A"></span>}--(0.5*\bigradius,\bigradius);

    \draw (0,0) -- (0.5*\bigradius,\bigradius) node[xshift=-20pt,yshift=-20pt]{<span data-mx-katex="R"></span>};
    \draw (0,0) -- (2*\angle:\littleradius) node[xshift=0pt,yshift=-8pt]{<span data-mx-katex="%5Cdelta"></span>};
    \node[xshift=-10pt,yshift=10pt] at (-\bigradius,0){<span data-mx-katex="C_R"></span>};
    \node[xshift=-10pt,yshift=10pt] at (-\littleradius,0){<span data-mx-katex="C_%7B%5Cdelta%7D"></span>};

    \draw[->](1.5,3.6)--(0.2,3.6);
    \node at (0.8,3.9){<span data-mx-katex="L_1"></span>};
    \draw[->](0.2,-3.6)--(1.5,-3.6);
    \node at (0.8,-3.9){<span data-mx-katex="L_2"></span>};
    \node at(-2,2.1){<span data-mx-katex="%5CGamma_1"></span>};
    \node at(-2,-2.1){<span data-mx-katex="%5CGamma_2"></span>};
    \node at(-2,0.5){<span data-mx-katex="%5Clambda_1"></span>};
    \node at(-2,-0.5){<span data-mx-katex="%5Clambda_2"></span>};
    \node at(2,0.5){<span data-mx-katex="L"></span>};
    % pole
    %\fill (0,0)  circle (3pt) ;
    %\draw[-Latex] (0.6,-0.8) node[right]{pole} --(0.15,-0.15);

    % axis
    \draw[-Latex] (-1.8*\radius,0) -- (1.3*\radius,0) node[below]{<span data-mx-katex="x"></span>} ;
    \draw[-Latex] (0,-1.2*\radius) -- (0,1.2*\radius) node[right]{<span data-mx-katex="iy"></span>};
    \end{tikzpicture}
\end{center}

\begin{center}
      \begin{tikzpicture}[>={Stealth[length=0.3cm,bend]},
                decoration={markings,
                mark= at position .1 with {\arrow{>}},
                mark= at position .32 with {\arrow{>}},
                mark= at position .55 with {\arrow{>}},
                % mark= at position 0.86 with {\arrow{>}},
                }
            ]
                \def\gap{0.3}
                \def\bigradius{3}
                \def\littleradius{0.5}
            \filldraw[postaction = {decorate}, thick ,fill=gray!40]
            let
                \n1 = {asin(\gap/2/\bigradius)},
                \n2 = {asin(\gap/2/\littleradius)}
            in
                (0+\n1:\bigradius) node[above right]{<span data-mx-katex="R"></span>}  arc (0+\n1:360-\n1:\bigradius) node[below left]{<span data-mx-katex="C_%7BR%7D"></span>} -- (0-\n2:\littleradius) arc (360-\n2:0+\n2:\littleradius) node[above right]{<span data-mx-katex="%5Cdelta"></span>} -- cycle;
                \draw[thick,->] (300:\littleradius) arc (300:130:\littleradius) node[above]{<span data-mx-katex="C_%7B%5Cdelta%7D"></span>};
            %\fill (-1.2,0)  circle (3pt) ;
            %\draw[->] (-1.2,-1) node[right]{pole} --(-1.2,-0.15);  % <--
            \draw[-Latex] (-1.5*\bigradius,0) -- (1.5*\bigradius,0) node[below]{<span data-mx-katex="x"></span>} ;
            \draw[-Latex] (0,-1.2*\bigradius) -- (0,1.2*\bigradius) node[right]{<span data-mx-katex="iy"></span>};
      \end{tikzpicture}
\end{center}