interior

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11: 2.1 Definitions and Elementary Properties

… ►For example, if

f(z)

is analytic for all sufficiently large

|z|

in a sector

\mathbf{S}

and

f(z)=O\left(z^{\nu}\right)

z\to\infty

\mathbf{S}

\nu

being real, then

f^{\prime}(z)=O\left(z^{\nu-1}\right)

z\to\infty

in any closed sector properly interior to

\mathbf{S}

and with the same vertex (Ritt’s theorem). …

12: 1.15 Summability Methods

… ►can be extended to the interior of the unit circle as an analytic function …

On the strip $a\leq\Re z\leq n$ , $f(z)$ is analytic in its interior, $f^{(2m)}(z)$ is continuous on its closure, and $f(z)=o\left(e^{2\pi|\Im z|}\right)$ as $\Im z\to\pm\infty$ , uniformly with respect to $\Re z\in[a,n]$ .

…

14: 19.2 Definitions

… ►The integral for

E\left(\phi,k\right)

is well defined if

k^{2}={\sin}^{2}\phi=1

, and the Cauchy principal value (§1.4(v)) of

\Pi\left(\phi,\alpha^{2},k\right)

is taken if

1-\alpha^{2}{\sin}^{2}\phi

vanishes at an interior point of the integration path. …

15: 18.18 Sums

… ►when

z

lies in the interior of

E

. …