# what do these complex analysis terms mean?

I'd like to know the definition of the following terms I'm not familiar with... and to know if my understanding is misguided or in the right direction. In the brackets is what I believe it to be - Thanks!

• Neighbourhood: (Essentially a small 'area' around, when referring to, a point that is not necessarily small.)

• Holomorphic (A term used to describe a function that is continuous, and is differentiable at every point in the domain and co-domain. What should immediately come to mind when I read this term?)

• Open set (A set that is has no 'gaps' within it (no more than an educated guess.))

• Analytic (Another, synonymous, term to holomorphic.)

• Disk (A $2$-dimensional space? - Please explain how one defines a disk.)

• Ball (A $3$D space.... what is an 'open' ball or a 'closed' ball?)

• Simple (when they say a simple closed path... what does this mean?)

Just revising (complex) analysis from the start of the year, so any help is much appreciated,

Once again, thanks!!

EDIT: About text books / Wikipedia... I can't actually ask it or them a question if I don't understand what they're saying, which most of the time I don't... Believe it or not, I have heard of wikipedia and it is always my first place to go look, but I'm after a simple definition from someone, who I can ask questions about if I need to. If you don't want to help me, then you don't have to answer at all!

• About 6/7 terms are used in analysis other than complex not to sure if holomorphic is strictly complex though. That is to say, if you are doing complex analysis, you should have been exposed to real analysis and encountered most of these terms already. – dustin Feb 18 '15 at 19:03
• Note that "analytic" and "holomorphic" are equivalent, but they are not really synonymous. "Holomorphic" means the complex derivative exists; "analytic" means the power series expansion exists. It is a remarkable fact of complex analysis that the two are equivalent. – Ian Feb 18 '15 at 19:05

## 1 Answer

• Neighbourhood of $p$: An set $N$ which constains an open set $U$ with $p \in U$,
• Holomorphic: complex differentiable at every point,
• Open set: A set $U$ such that for any $p \in U$ there is an $\epsilon>0$, such that the ball $B_\epsilon(p) \subseteq U$,
• Analytic function: a function that is locally given by a convergent power series,
• (closed) Disk: $D_r(p) := \{x \in \mathbb{R}^2 :\, |x-p| \leq r\}$,
• Ball: $B_r(p) := \{x \in X :\, |x-p| < r\}$,
• simple curve: a curve that is an injective embedding.
• Most authors consider disks to be either open or closed (specifying one way or the other). – Ian Feb 18 '15 at 19:04
• @Ian ok, changed my answer to indicate disk usually means a region in the euclidean plane bounded by a circle – Loreno Heer Feb 18 '15 at 19:07
• Thanks for answering. what's the difference between differentiable at every point and COMPLEX differentiable at every point? And what exactly do you mean by a set N which 'constrains an open set U with p in U? Thanks again – Douglas Feb 18 '15 at 19:12
• Also, could you give a simple example of a set that wouldn't be open? it seems to me every set could have a point with a ball around it of a small radius and still remain within the set. Thanks – Douglas Feb 18 '15 at 19:13
• @Douglas For example, If you draw a ball of radius $2$ around $(0,0)$, it contains the point $(\sqrt{2},\sqrt{2})$, which doesn't have rational coordinates. – Ian Feb 18 '15 at 20:27