Meromorphic functions: Difference between revisions
imported>Dmitrii Kouznetsov (New page: In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the...) |
imported>Meg Taylor No edit summary |
||
| (2 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
{{subpages}} | |||
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are [[pole]]s for the function. (The terminology comes from the Ancient Greek “meros” (μέρος), meaning part, as opposed to “holos” (ὅλος), meaning whole.) | In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are [[pole]]s for the function. (The terminology comes from the Ancient Greek “meros” (μέρος), meaning part, as opposed to “holos” (ὅλος), meaning whole.) | ||
| Line 5: | Line 6: | ||
Many of [[special function]]s are meromorphic. | Many of [[special function]]s are meromorphic. | ||
For example, [[factorial]] is meromorphic in the whole | For example, [[factorial]] is meromorphic in the whole complex plane (it has [[countable set]] of poles at the negative integer values of the argument), but [[logarithm]] is not, because it has cutline at the negative part of the real axis. However, the same logarithm becomes | ||
meromorphic being considered on the domain of numbers | meromorphic being considered on the domain of numbers with positive real part. | ||
Latest revision as of 10:05, 10 October 2013
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. (The terminology comes from the Ancient Greek “meros” (μέρος), meaning part, as opposed to “holos” (ὅλος), meaning whole.)
In particular, every holomorphic function can be considered as meromorphic.
Many of special functions are meromorphic.
For example, factorial is meromorphic in the whole complex plane (it has countable set of poles at the negative integer values of the argument), but logarithm is not, because it has cutline at the negative part of the real axis. However, the same logarithm becomes meromorphic being considered on the domain of numbers with positive real part.