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Results tagged with ca.classical-analysis-and-odes
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user 125758
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
5
votes
1
answer
314
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Implicit function theorem with singularities of any order
Let $\mathcal{U}\subset \mathbb{R}\times \mathbb{C}$ a neighborhood of $(0,0)$, and $f:\mathcal{U}\to \mathbb{C}$ differentiable in the first variable and holomorphic in the second variable, with $f(0 …
- 211
1
vote
Accepted
Implicit function theorem with singularities of any order
Rouché's theorem provides a simple solution.
Let $f_1(x,z)=z^k+x^h$ and $f_2(x,z)=h.o.t.$. For each $x$, the functions $f_1$ and $f_2$ are holomorphic in $z$. We know $f_2(x,z)=o(z^k)+o(x^h)$, so ther …
- 211