a a above. After After also analytic and and and and and and and and and and and an'

any any As as as assume assume at at

be because by by

$\mathrm{\bigvee }\mathrm{\cap }\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}$ complex component component con- contains coordinates Corollary Corollary

nurve

Denote

aigenvalue Estimates estimates extension

following following follows For for for for for form Francine from functions

$\mathrm{\to }\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}$, generality) get

has has have hence Henri-Michel $\mathit{h}\mathit{o}\mathit{l}\mathit{d}{\mathrm{(}}^{\mathrm{2}}\mathrm{)}$: holomorphic holomorphic hypotheses

[ $\mathrm{n}$ in in into Introducing is is is

know

Let let Let Levi lies line loss loss

Maire make may means Meylan

$\mathrm{(}\mathrm{\prime }\mathrm{o}\mathrm{,}\mathrm{}{\mathit{z}}_{\mathrm{2}}\mathrm{,}\mathrm{0}\mathrm{)}\mathrm{\in }\mathit{M}\mathrm{\left|}\mathrm{\right(}\mathit{z}\mathrm{,}\mathrm{}\mathit{s}\mathrm{+}\mathrm{i}\mathit{t}\mathrm{\left)}\mathrm{\right|}{\mathit{Z}}_{\mathrm{2}}\mathrm{=}{\mathit{z}}_{\mathrm{2}}\mathrm{,}$ $\mathit{W}\mathrm{=}\mathit{w}$. $\frac{\mathrm{\partial }\mathit{g}}{\mathrm{\partial }\mathit{s}}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{>}\mathrm{0}\mathrm{,}$ $\mathrm{(}\mathrm{\partial }\mathit{g}\mathrm{/}\mathrm{\partial }\mathit{s}\mathrm{)}\mathrm{\left(}\mathrm{0}\mathrm{\right)}$. ${\mathit{F}}_{\mathrm{1}}\mathrm{(}\mathrm{0}\mathrm{,}\mathrm{}{\mathit{Z}}_{\mathrm{2}}\mathrm{,}\mathrm{0}\mathrm{)}\mathrm{\equiv }\mathrm{0}\mathrm{(}\mathrm{\partial }\mathit{G}\mathrm{/}\mathrm{\partial }\mathit{w}\mathrm{)}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{\ne }\mathrm{0}$

${\mathit{Z}}_{\mathrm{1}}\mathrm{=}{\mathit{z}}_{\mathrm{1}}\mathrm{-}{\mathit{C}}_{\mathrm{1}}\mathrm{\left(}{\mathit{z}}_{\mathrm{2}}\mathrm{\right)}\mathrm{,}$ $\mathrm{\left\{}\mathrm{\right(}\mathrm{0}\mathrm{,}\mathrm{}{\mathit{Z}}_{\mathrm{2}}\mathrm{,}\mathrm{0}\mathrm{\left)}\mathrm{\right|}{\mathit{Z}}_{\mathrm{2}}\mathrm{\in }\mathrm{C}\mathrm{\}}{\mathit{z}}_{\mathrm{2}}\mathrm{\mapsto }{\mathrm{\partial }}_{\mathit{z}}^{\mathit{\alpha }}{G}\mathrm{(}\mathrm{0}\mathrm{,}\mathrm{}{\mathit{z}}_{\mathrm{2}}\mathrm{,}\mathrm{0}\mathrm{\left)}\mathit{u}\mathrm{\right(}\mathit{z}\mathrm{,}\mathit{w}\mathrm{)}\mathrm{/}\mathit{v}\mathrm{(}\mathit{z}\mathrm{,}\mathit{w}\mathrm{)}\mathrm{\to }\mathrm{0}\mathrm{,}$ $\mathit{u}\mathrm{(}\mathit{z}\mathrm{,}\mathit{w}\mathrm{)}\mathrm{=}\mathit{o}\mathrm{\left(}\mathit{v}\mathrm{\right(}\mathit{z}\mathrm{,}\mathit{w}\mathrm{\left)}\mathrm{\right)}$

$\mathrm{[}\mathrm{Im}\mathit{w}\mathrm{>}\mathit{\phi }\mathrm{(}\mathit{z}\mathrm{,}\overline{\mathit{z}}\mathrm{,}\mathrm{}\mathrm{Re}\mathit{w}\mathrm{\left)}\mathrm{\right\}}$. $\mathit{\phi }\mathrm{(}\mathrm{0}\mathrm{,}\mathrm{}{\mathit{Z}}_{\mathrm{2}}\mathrm{,}\mathrm{0}\mathrm{,}{\overline{\mathit{Z}}}_{\mathrm{2}}\mathrm{,}\mathrm{0}\mathrm{)}\mathrm{\equiv }\mathrm{0}$.

$\mathrm{)}$ ${G}\mathit{g}\mathit{H}{H}\mathit{h}\mathit{M}\mathit{M}\mathit{M}\mathrm{\Omega }\mathrm{\square }{\mathit{C}}^{\mathrm{\infty }}{\mathit{M}}^{\mathrm{\prime }}\mathrm{\Omega }\mathrm{\cap }\overline{\mathit{\phi }}{\mathit{z}}_{\mathrm{2}}\mathit{\beta }\mathrm{\ne }\mathrm{0}\mathit{h}\mathrm{,}$ $\mathit{M}\mathit{t}\mathrm{\ge }\mathrm{0}\mathrm{0}\mathrm{\in }{\mathrm{C}}^{\mathrm{3}}\mathit{k}\mathrm{\ne }\mathrm{0}\mathrm{,}{\overline{\mathit{F}}}_{\mathrm{1}}$. $\mathit{j}$, A $\mathrm{\in }\mathrm{N}$

$\overline{\mathit{\rho }}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathit{\alpha }\mathrm{,}$ $\mathit{\beta }\mathrm{\in }{\mathrm{N}}^{\mathrm{2}}\overline{\mathit{\phi }}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{>}\mathrm{0}\overline{\mathit{\phi }}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{\ne }\mathrm{0}\mathit{h}\mathrm{,}$ $\mathit{M}\mathrm{,}$ ${\mathit{M}}^{\mathrm{\prime }}$, (; $\mathrm{(}\mathit{z}\mathrm{,}\mathrm{}\mathit{w}\mathrm{)}\mathrm{\to }\mathrm{O}\mathrm{,}$ $\mathrm{(}\mathit{z}\mathrm{,}\mathit{w}\mathrm{)}\mathrm{\to }\mathrm{0}$.

${\mathrm{\partial }}_{\mathit{z}}^{\mathit{\alpha }}{\mathrm{\partial }}_{\overline{\mathit{z}}}^{\mathit{\beta }}{\mathrm{\partial }}_{\mathit{w}}^{\mathit{j}}\mathrm{\partial }\frac{\mathit{k}}{\mathit{w}}{G}{\mathrm{|}}_{\mathit{M}}\mathrm{=}\mathrm{0}$. $\mathit{\phi }\mathrm{(}\mathit{z}\mathrm{,}\overline{\mathit{z}}\mathrm{,}\mathrm{0}\mathrm{)}\mathrm{=}\mathrm{\left|}{\mathit{z}}_{\mathrm{1}}{\mathrm{|}}^{\mathrm{2}}\overline{\mathit{\phi }}\mathrm{\right(}\mathit{z}\mathrm{,}\overline{\mathit{z}}\mathrm{\left)}{\mathit{F}}_{\mathrm{1}}\mathrm{\right(}{\mathit{Z}}_{\mathrm{1}}\mathrm{,}{\mathit{Z}}_{\mathrm{2}}\mathrm{,}\mathrm{0}\mathrm{)}\mathrm{=}{\mathit{Z}}_{\mathrm{1}}{\overline{\mathit{F}}}_{\mathrm{1}}\mathrm{(}{\mathit{Z}}_{\mathrm{1}}\mathrm{,}{\mathit{Z}}_{\mathrm{2}}\mathrm{)}$

$\mathit{g}\mathrm{(}\mathit{z}\mathrm{,}\overline{\mathit{z}}\mathrm{,}\mathrm{}\mathit{s}\mathrm{+}\mathrm{i}\mathit{t}\mathrm{)}\mathrm{|}\mathrm{\ge }\frac{\mathrm{1}}{\mathrm{8}}\frac{\mathrm{\partial }\mathit{g}}{\mathrm{\partial }\mathit{s}}\mathrm{(}\mathrm{0}\mathrm{\left)}\mathrm{\right|}\mathit{s}\mathrm{+}\mathrm{i}\mathit{t}\mathrm{\left|}{G}\mathrm{\right(}\mathit{z}\mathrm{,}\mathrm{}\mathit{w}\mathrm{)}\mathrm{=}\mathit{o}\mathrm{(}\mathrm{\left|}{\mathit{z}}_{\mathrm{1}}{\mathrm{|}}^{\mathrm{2}}\mathrm{\right)}\mathrm{+}\mathit{w}\frac{\mathrm{\partial }{G}}{\mathrm{\partial }\mathit{w}}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{+}\mathit{o}\mathrm{\left(}\mathrm{\right|}\mathit{w}\mathrm{\left|}\mathrm{\right)}$,

$\mathit{q}\mathrm{(}\mathit{z}\mathrm{,}\overline{\mathit{z}}\mathrm{,}\mathrm{}\mathit{s}\mathrm{+}\mathrm{i}\mathit{t}\mathrm{)}\mathrm{=}{G}\mathrm{(}\mathit{z}\mathrm{,}\overline{\mathit{z}}\mathrm{,}\mathrm{}\mathit{s}\mathrm{+}\mathrm{i}\mathit{t}\mathrm{+}\mathrm{i}\mathit{\phi }\mathrm{(}\mathit{z}\mathrm{,}\overline{\mathit{z}}\mathrm{,}\mathrm{}\mathit{s}\mathrm{+}\mathrm{i}\mathit{t}\mathrm{\left)}\mathrm{\right)}$.

neighborhood nonzero

obtain of of of of of of of of of of or or orem

3. 192 1.3 2. 2.1, 2.2 3.2). 4. 4.1. (1.1) (1.1), () $\mathrm{(}\mathrm{3}\mathrm{.}\mathrm{3}\mathrm{)}$. $\mathrm{(}\mathrm{3}\mathrm{.}\mathrm{4}\mathrm{)}$. $\mathrm{(}\mathrm{4}\mathrm{.}\mathrm{1}\mathrm{)}\mathrm{(}\mathrm{4}\mathrm{.}\mathrm{2}\mathrm{)}\mathrm{(}\mathrm{4}\mathrm{.}\mathrm{3}\mathrm{)}\mathrm{(}\mathrm{4}\mathrm{.}\mathrm{4}\mathrm{)}$ (cf.

$\mathrm{(}\mathrm{\prime }\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{t}$

paragraph power Proof. Proof. Proposition Proposition

question

real Remark

$\mathrm{\supset }\mathrm{a}\mathrm{⌝}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{N}$ see series sign sign small, small. some suppose

that that that that that that The the the the the the the the the the the the The-

Then Then therefore Therefore, third this this tinuity to transversal transversal

$\mathrm{t}\mathrm{1}\mathrm{S}$ using using usual,

We we we we we we we we where which with Wrthont