images of fixed point continued fractions
DESCRIPTION
Simple topographical images of several fixed point continued fractions.TRANSCRIPT
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Classroom Notes: Images of Fixed-Point Continued Fractions
John Gill December 2013
Abstract: Computer generated images in the complex plane of iterations of certain analytic continued fractions.
Fixed-point continued fractions have the form
(1) 1 1 2 2
1 1 2 2
+ + where ,k k are the fixed points of the function
( ) k kkk k
f
= + . The nth approximant of the continued fraction then can be written
1 2( ) ( )
n nF f f f = . If ( )
k kz = and ( )
k kz = , we have
1 2( , ) ( , )
n nF z f f f z = . The following images show magnitudes of simple complex flux (SCF), ( ,0)nF z , or fixed-point flux (FPF), ( ,0)nF z z . Dark=small, light=large. Several
variations on FPCFs are shown, as well. [Liberty Basic V 4.04 (2013)]
Example 1: ( , )kkz
f zk z
= + , 6 , 6x y , 10n = SCF & FPF
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Example 2: ( )21
( , )1
k
ik zf z
ik z
+=
+ + and
( )2.5( , )
.5k
ik zf z
ik z
+=
+ + , 6 , 6x y , 10n = SCF
A Variation on fixed-point CFs. Note: a programming error produced these two images!
Example 3: (1 )
( , )1
k
kz kzf z
=
, 6 , 6x y , 10n = SCF
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Example 4: (1 )
( , )
1 k
k
k
kz kzf z
z
z
=
+
, =1+ik k
, 6 , 6x y , 10n = SCF
Example 5: ( )1
( , )1
k
ik zf z
ik z
=
+ , 7 , 7x y , 10n = SCF
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Example 6: ( , )z
f zk z
= + , 6 , 6x y , 10n = SCF. Another variation on FPCFs.
Example 7: 2
( , )kkz
f zk z
= + , 10 , 10x y , 10n = SCF. Another variation on FPCFs.
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Example 8:2(1 2 )
( , )1 2
ki zf z
ki z
+=
+ + , 10 , 10x y , 10n = SCF. Another variation on
FPCFs.
Example 9: (1 )
( , )1
i ik zf z
ik z
+=
+ + , 8 , 8x y , 10n = SCF. Variation on FPCFs.