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The exponential function omits exactly zero and shows little Picard is sharp
Example
The exponential function is entire, omits exactly the value , and therefore shows that Little Picard's "at most one omitted finite value" is sharp.
Facts & Assumptions
Given: The complex exponential function.
The exponential is entire (The complex exponential is entire and its complex derivative is itself).
Its image is exactly (The complex exponential maps onto ).
Little Picard allows at most one omitted finite value (Little Picard theorem).
Verification
Facts [L1] and [L2] show that is a nonconstant entire function whose omitted finite-value set is exactly .
Fact [L3] says no nonconstant entire function can omit two finite values, while step 1.1 exhibits one omitting exactly one. Therefore the bound in Little Picard is sharp.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Aleksander Simonic, The Ahlfors lemma and Picard's theorems, §6.2 (standard reference, not scraped)