Hartog's theorems

Hartogs' Theorem (Separate Holomorphy ⇒ Joint Holomorphy)

If a function $f:\mathrm{\Omega }\subset {\mathbb{C}}^n\to \mathbb{C}$ is holomorphic in each variable separately, then $f$ is holomorphic in all variables jointly (for $n\ge 2$).

Hartogs' Extension Theorem

If $f$ is holomorphic on $\mathrm{\Omega }\setminus K$, where $K\subset \mathrm{\Omega }$ is compact and $\mathrm{\Omega }\setminus K$ is connected, then $f$ extends holomorphically over $K$.

The "Magic"

Both results reflect the rigidity of holomorphic functions in several complex variables:

• No isolated singularities (unlike the case of one complex variable).
• Singular sets must be large (e.g., cannot be just points or small compact sets).