# Non-singular boundary point

*regular boundary point*

An accessible boundary point (cf. Attainable boundary point) $ \zeta $ of the domain of definition $ D $ of a single-valued analytic function $ f ( z) $ of a complex variable $ z $ such that $ f ( z ) $ has an analytic continuation to $ \zeta $ along any path inside $ D $ to $ \zeta $. In other words, a non-singular boundary point is accessible, but not singular. See also Singular point of an analytic function.

#### Comments

Note that the same point in the boundary of $ D $ may give rise to several different accessible boundary points, some of which may be singular, others regular. E.g., consider the domain $ D = \mathbf C \setminus ( - \infty , 0 ] $, and the function $ f ( z) = ( h ( z) - \pi i ) ^ {-} 1 $, where $ h $ is the principal value of $ \mathop{\rm log} z $. Then "above" $ - 1 $ there are two accessible boundary points: one singular, corresponding to approach along $ z = - 1 + i t $, $ 0 \leq t \leq 1 $; one regular, corresponding to approach along $ z = - 1 - i t $, $ 0 \leq t \leq 1 $.

#### References

[a1] | A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. Chapts. 2; 8 (Translated from Russian) |

**How to Cite This Entry:**

Non-singular boundary point.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Non-singular_boundary_point&oldid=48003