two_dim_delta.saty

@require: stdjabook
@require: code
@require: itemize
@require: tabular
@require: math
@import: local

document (|
  title = {2次元極座標のデルタ};
  author = {hsjoihs};
  show-title = true;
  show-toc = false;
|) '<
  +section{}<
    +math(${
      x \vec = \paren{x, y}, x_0 \vec = \paren{x_0, y_0}
    });
    +math(${
      \app{\delta}{x \vec - x_0 \vec }  \equiv \app{\delta}{x-x_0}\app{\delta}{y-y_0}
    });
    +math(${
      x = r\cos\theta, y = r\sin\theta
    });
    +math(${
      x_0 = r_0\cos\theta_0, y_0 = r_0\sin\theta_0
    });
    +math(${
      \app{\delta}{x \vec - x_0 \vec} = \lim_{n \arrow \infty} \frac{n}{\pi} e^{-n\paren{x-x_0}^2-n\paren{y-y_0}^2}
    });
    +math(${
      \paren{x-x_0}^2+\paren{y-y_0}^2 = x^2 + y^2 - 2xx_0-2yy_0 +x_0^2 +y_0^2 
    });
    +math(${
      = r^2 + r_0^2 - 2\paren{x,y}\cdot\paren{x_0,y_0}
      = r^2 + r_0^2 - 2rr_0 \app{\cos}{\theta -\theta _0}
    });
    +math(${
      \app{\delta}{x \vec - x_0\vec}
      = \lim_{n \arrow \infty} \frac{n}{\pi} e^{-n\paren{x-x_0}^2-n\paren{y-y_0}^2}
      = \lim_{n \arrow \infty} \frac{n}{\pi}  e^{-n\paren{r^2 + r_0^2 - 2rr_0 \app{\cos}{\theta -\theta_0}}}
    });
    +pn{
      どうせ高次の項など効いてこないので（そういうとこやぞ）
    }
    +math(${
      = \lim_{n \arrow \infty} \frac{n}{\pi}  
      e^{
        -n\paren{
          r^2 + r_0^2 - 2rr_0 \paren{
            1-\frac{1}{2}\paren{\theta -\theta_0}^2
          }
        }
      } 
    });
    +math(${
      = \lim_{n \arrow \infty} \frac{n}{\pi}  
      e^{
        -n\paren{
          r^2 + r_0^2 - 2rr_0 + rr_0 \paren{
            \theta -\theta_0
          }^2
        }
      }
    });
    +math(${
      = \lim_{n \arrow \infty}  \sqrt{\frac{n}{\pi}}  
      e^{
        -n\paren{r- r_0}^2
      } \cdot \sqrt{\frac{n}{\pi}} 
      e^{ 
        -n \paren{
          rr_0\paren{\theta -\theta_0}^2
        }
      } 
    });
    +math(${
      =\app{\delta}{r-r_0}
      \app{\delta}{
        \sqrt{rr_0}\paren{\theta -\theta_0}
      }
    });
    +math(${
      \int \int \app{f}{r, \theta} \app{\delta}{x\vec - x_0\vec} dr d\theta =
      \int \int \app{f}{r, \theta} \app{\delta}{r-r_0}
      \app{\delta}{
        \sqrt{rr_0} \paren{\theta -\theta_0}
      } dr d\theta
    });
    +math(${
      = \int \app{f}{r_0, \theta} \app{\delta}{
        \sqrt{r_0r_0} \paren{\theta -\theta_0}
      }  d\theta
    });
    +math(${
      = \int \app{f}{r_0, \theta} \frac{1}{r_0}
      \app{\delta}{\theta -\theta_0 } d\theta
    });
    +math(${
      = \int \int \app{f}{r, \theta}  \frac{1}{r} 
      \app{\delta}{r-r_0}
      \app{\delta}{\theta -\theta_0}
      dr d\theta
    });
    +math(${
      \app{\delta}{x \vec - x_0 \vec} = \frac{1}{r} 
      \app{\delta}{r-r_0}
      \app{\delta}{\theta -\theta_0}
    });
  >
>