Essentially special point

Isolated Point $z_{0}$ ${\ displaystyle z_ {0}}$ $z _ {{0}}$ the functions $f(z)$ ${\ displaystyle f (z)}$ $f (z)$ holomorphic in some punctured neighborhood of this point is called essentially singular if the limit ${\textstyle \lim _{z\to {z_{0}}}f(z)}$ ${\ textstyle \ lim _ {z \ to {z_ {0}}} f (z)}$ ${\ textstyle \ lim _ {z \ to {z_ {0}}} f (z)}$ does not exist.

Content

Essentially Feature Point Criterion

Point $z_{0}$ ${\ displaystyle z_ {0}}$ is an essential feature point of a function $f(z)$ ${\ displaystyle f (z)}$ if and only if in the expansion of the function $f(z)$ ${\ displaystyle f (z)}$ in a row of Laurent in a punctured neighborhood of a point $z_{0}$ ${\ displaystyle z_ {0}}$ the main part contains an infinite number of nonzero terms, i.e., in the expansion $f(z)=\sum _{k=-\infty }^{\infty }{f_{k}}(z-z_{0})^{k}$ ${\ displaystyle f (z) = \ sum _ {k = - \ infty} ^ {\ infty} {f_ {k}} (z-z_ {0}) ^ {k}}$ number of factors $f_{k}\neq 0$ ${\ displaystyle f_ {k} \ neq 0}$ , $k<0$ ${\ displaystyle k <0}$ endlessly.

Sokhotsky – Weierstrass Theorem

Whatever complex number $B$ ${\ displaystyle B}$ , for anyone $\varepsilon >0$ ${\ displaystyle \ varepsilon> 0}$ in any neighborhood of an essentially singular point $z_{0}$ ${\ displaystyle z_ {0}}$ there is a point $z$ ${\ displaystyle z}$ such that $|f(z)-B|<\varepsilon$ ${\ displaystyle | f (z) -B | <\ varepsilon}$ .

Literature

Bitsadze A.V. Fundamentals of the theory of analytic functions of a complex variable - M., Science, 1969.
Shabbat B.V., Introduction to complex analysis - M., Science, 1969.

Essentially special point

Content

Essentially Feature Point Criterion

Sokhotsky – Weierstrass Theorem

See also

Literature

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