ワイエルシュトラスの因数分解定理
( 1 − z ) = exp  ( log  ( 1 − z ) ) = exp  ( − h ∞ ( z ) ) {\displaystyle (1-z)=\exp(\log(1-z))=\exp \left(-h_{\infty }(z)\right)}
1 1 − z = exp  ( h ∞ ( z ) ) {\displaystyle {\tfrac {1}{1-z}}=\exp \left(h_{\infty }(z)\right)}
である。
また、 h n ( z ) {\displaystyle h_{n}(z)} を微分すると、 h n ′ ( z ) = 1 + z 1 + z 2 + ⋯ + z n − 1 = 1 − z n 1 − z {\displaystyle h'_{n}(z)=1+z^{1}+z^{2}+\cdots +z^{n-1}={\tfrac {1-z^{n}}{1-z}}} となる。
r n ( z ) = h ∞ ( z ) − h n ( z ) {\displaystyle r_{n}(z)=h_{\infty }(z)-h_{n}(z)}
と定義すれば、
E n ( z ) = ( 1 − z ) exp  ( h n ( z ) ) = ( 1 − z ) exp  ( h ∞ ( z ) − r n ( z ) ) {\displaystyle E_{n}(z)=(1-z)\exp(h_{n}(z))=(1-z)\exp(h_{\infty }(z)-r_{n}(z))} = ( 1 − z ) 1 1 − z exp  ( − r n ( z ) ) = exp  ( − r n ( z ) ) {\displaystyle =(1-z){\tfrac {1}{1-z}}\exp \left(-r_{n}(z)\right)=\exp \left(-r_{n}(z)\right)}