A Theorem on the Convergence of a Continued Fraction Page: 4
iii, 37 leavesView a full description of this thesis.
Extracted Text
The following text was automatically extracted from the image on this page using optical character recognition software:
4
- {H <l}| C R# or|w - c|^r. Therefore, If Z la In the
cirele JZ - Qjf; R, then w is in the cirelejw - «j^p#
If jw — ej^.r til on. since w 9 Z +H# a * Q 4 H# and p * 8,
we haw | (%■+ S) - (Q + H)| £ R, ©r|z - a|i .
theorem 1 |Z «• R and w Z -f*H than there exiata
a number © and a real number r such that |w - ©j>i*# also if
\w - e|^ r, then|Z - <i|e: R.
fhis can b© proved in the s«m® method as used in
Theorem 1*4.
ls@—If w Z4 H, a /O and aZ+ aZ 2p then there
exists a number § and a real number % such that cw+ cW ±.2q,
Furthermore if cw + cw^2q then aZ + aZ^ 2p.
X*t ¥ w* a « a, and ^i¥H) + p * q. Since w = Z + H#
then Z w - H and Z m m • S# If aZf aZ£J|p# then
W jw ®]4 a \w H^£ 2p$
¥w - SH 4. aW - aH <: 2pj
aw+aw - (IS all) .f 8p*
aw-f-aw • 2 M, (aft) £ 2p;
I* +al± aj^ (aH)t^J
Therefor® if Z is in the half plane iZ+ai^2p then w is
in the half plan® i 4 cSr f:8q.
If ew4- cw f Bt, then since w « Z -Hi, w I ft, c = a,
e = a, and q (aH)+- p=fl-we have
I [Z+H]+ B ^z + gji Bfa(aB) + J,
Upcoming Pages
Here’s what’s next.
Search Inside
This thesis can be searched. Note: Results may vary based on the legibility of text within the document.
Tools / Downloads
Get a copy of this page or view the extracted text.
Citing and Sharing
Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.
Reference the current page of this Thesis.
Kostelec, John C. A Theorem on the Convergence of a Continued Fraction, thesis, January 1953; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc107838/m1/7/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .