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== Bimodular approximants ==

If ''x'' is near to 1, then ln(''x'')/2 is approximated by {{nowrap| bim(''x'') {{=}} (''x'' - 1)/(''x'' + 1) }}, the bimodular approximant function, which is the {{w|Padé approximant}} of order (1, 1) to ln(''x'')/2 near 1. The bimodular approximant function is a {{w|Möbius transformation}} and hence has an inverse, which we denote {{nowrap| mib(''x'') {{=}} (1 + ''x'')/(1 - ''x'') }}, which is the (1, 1) Padé approximant around 0 for exp(2''x''). Then {{nowrap| bim(exp(2''x'')) {{=}} tanh(''x'') }}, and therefore {{nowrap| ln(mib(''x''))/2 {{=}} artanh(''x'') {{=}} ''x'' + ''x''3/''x'' + ''x''5/5 + … }}, from which it is apparent that bim(''x'') approximates ln(''x'')/2, and mib(''x'') approximates exp(2''x''), to the second order; we may draw the same conclusion by directly comparing the series for exp(2''x'') = 1 + 2''x'' + 2''x''2 + O(''x''3) with {{nowrap| mib(''x'') {{=}} 1 + 2''x'' + 2''x''2 + O(''x''3) }} and {{nowrap| ln(''x'')/2 {{=}} (''x'' - 1)/2 - (''x'' - 1)2/4 + O(''x''3) }}, which is the same to the second order as bim(''x''). Using mib, we may also define {{nowrap| BMC(''a'', ''b'') = DPC(mib(''a''), mib(''b'')) }}, where BMC is an acronym for ''bimodular comma''.

If ''x'' is near to 1, then ln(''x'')/2 is approximated by {{nowrap| bim(''x'') {{=}} (''x'' - 1)/(''x'' + 1) }}, the bimodular approximant function, which is the {{w|Padé approximant}} of order (1, 1) to ln(''x'')/2 near 1. The [[Logarithmic approximants|bimodular approximant]] function is a {{w|Möbius transformation}} and hence has an inverse, which we denote {{nowrap| mib(''x'') {{=}} (1 + ''x'')/(1 - ''x'') }}, which is the (1, 1) Padé approximant around 0 for exp(2''x''). Then {{nowrap| bim(exp(2''x'')) {{=}} tanh(''x'') }}, and therefore {{nowrap| ln(mib(''x''))/2 {{=}} artanh(''x'') {{=}} ''x'' + ''x''3/''x'' + ''x''5/5 + … }}, from which it is apparent that bim(''x'') approximates ln(''x'')/2, and mib(''x'') approximates exp(2''x''), to the second order; we may draw the same conclusion by directly comparing the series for exp(2''x'') = 1 + 2''x'' + 2''x''2 + O(''x''3) with {{nowrap| mib(''x'') {{=}} 1 + 2''x'' + 2''x''2 + O(''x''3) }} and {{nowrap| ln(''x'')/2 {{=}} (''x'' - 1)/2 - (''x'' - 1)2/4 + O(''x''3) }}, which is the same to the second order as bim(''x''). Using mib, we may also define {{nowrap| BMC(''a'', ''b'') = DPC(mib(''a''), mib(''b'')) }}, where BMC is an acronym for ''bimodular comma''.

If ''r'' is as above we have that {{nowrap| ''r'' {{=}} bim(''a'')/bim(''b'') }}, and depending on common factors the corresponding Don Page comma is equal to an ''n''-th power of {{nowrap| ''a''bim(''b'') / ''b''bim(''a'') {{=}} mib('''u''')'''v'''/mib('''v''')'''u''' }} for some ''n''. If we set ''a'' = 1 + ''x'', ''b'' = 1 + ''y'', then ''r'' = ''r''(''x'', ''y'') is an analytic function of two complex variables with a power series expansion around {{nowrap| ''x'' {{=}} 0 }}, {{nowrap| ''y'' {{=}} 0 }}. This expansion begins as ''r''(''x'', ''y'') = 1 - (''xy''3 - ''x''3''y'')/24 + (3''xy''4 + ''x''2''y''3 - ''x''3''y''2 - 3''x''4''y'')/48 + …, with its first nonconstant term of total degree four, and so when ''x'' and ''y'' are small, ''r''(''x'', ''y'') will be close to 1. The ''n''-th power will still be close to 1, and if there are common factors in the denominators of the exponents, the resulting comma will be less complex (lower in height), and this is the really interesting case. For example, if ''a'' = 7/6 and ''b'' = 27/25, we obtain (7/6)1/26/(27/25)1/13, the lcm of 13 and 26 is 26, and taking the 26th power gives us 4375/4374. The common factor of 13 leads to the resulting comma (4375/4374) being relatively simple.

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 == Bimodular approximants ==
 {{Todo|clarify|inline=1|text=rewrite GWS gibberish}}
-If ''x'' is near to 1, then ln(''x'')/2 is approximated by {{nowrap| bim(''x'') {{=}} (''x'' - 1)/(''x'' + 1) }}, the bimodular approximant function, which is the {{w|Padé approximant}} of order (1, 1) to ln(''x'')/2 near 1. The bimodular approximant function is a {{w|Möbius transformation}} and hence has an inverse, which we denote {{nowrap| mib(''x'') {{=}} (1 + ''x'')/(1 - ''x'') }}, which is the (1, 1) Padé approximant around 0 for exp(2''x''). Then {{nowrap| bim(exp(2''x'')) {{=}} tanh(''x'') }}, and therefore {{nowrap| ln(mib(''x''))/2 {{=}} artanh(''x'') {{=}} ''x'' + ''x''<sup>3</sup>/''x'' + ''x''<sup>5</sup>/5 + … }}, from which it is apparent that bim(''x'') approximates ln(''x'')/2, and mib(''x'') approximates exp(2''x''), to the second order; we may draw the same conclusion by directly comparing the series for exp(2''x'') = 1 + 2''x'' + 2''x''<sup>2</sup> + O(''x''<sup>3</sup>) with {{nowrap| mib(''x'') {{=}} 1 + 2''x'' + 2''x''<sup>2</sup> + O(''x''<sup>3</sup>) }} and {{nowrap| ln(''x'')/2 {{=}} (''x'' - 1)/2 - (''x'' - 1)<sup>2</sup>/4 + O(''x''<sup>3</sup>) }}, which is the same to the second order as bim(''x''). Using mib, we may also define {{nowrap| BMC(''a'', ''b'') = DPC(mib(''a''), mib(''b'')) }}, where BMC is an acronym for ''bimodular comma''.
+If ''x'' is near to 1, then ln(''x'')/2 is approximated by {{nowrap| bim(''x'') {{=}} (''x'' - 1)/(''x'' + 1) }}, the bimodular approximant function, which is the {{w|Padé approximant}} of order (1, 1) to ln(''x'')/2 near 1. The [[Logarithmic approximants|bimodular approximant]] function is a {{w|Möbius transformation}} and hence has an inverse, which we denote {{nowrap| mib(''x'') {{=}} (1 + ''x'')/(1 - ''x'') }}, which is the (1, 1) Padé approximant around 0 for exp(2''x''). Then {{nowrap| bim(exp(2''x'')) {{=}} tanh(''x'') }}, and therefore {{nowrap| ln(mib(''x''))/2 {{=}} artanh(''x'') {{=}} ''x'' + ''x''<sup>3</sup>/''x'' + ''x''<sup>5</sup>/5 + … }}, from which it is apparent that bim(''x'') approximates ln(''x'')/2, and mib(''x'') approximates exp(2''x''), to the second order; we may draw the same conclusion by directly comparing the series for exp(2''x'') = 1 + 2''x'' + 2''x''<sup>2</sup> + O(''x''<sup>3</sup>) with {{nowrap| mib(''x'') {{=}} 1 + 2''x'' + 2''x''<sup>2</sup> + O(''x''<sup>3</sup>) }} and {{nowrap| ln(''x'')/2 {{=}} (''x'' - 1)/2 - (''x'' - 1)<sup>2</sup>/4 + O(''x''<sup>3</sup>) }}, which is the same to the second order as bim(''x''). Using mib, we may also define {{nowrap| BMC(''a'', ''b'') = DPC(mib(''a''), mib(''b'')) }}, where BMC is an acronym for ''bimodular comma''.
 If ''r'' is as above we have that {{nowrap| ''r'' {{=}} bim(''a'')/bim(''b'') }}, and depending on common factors the corresponding Don Page comma is equal to an ''n''-th power of {{nowrap| ''a''<sup>bim(''b'')</sup> / ''b''<sup>bim(''a'')</sup> {{=}} mib('''u''')<sup>'''v'''</sup>/mib('''v''')<sup>'''u'''</sup> }} for some ''n''. If we set ''a'' = 1 + ''x'', ''b'' = 1 + ''y'', then ''r'' = ''r''(''x'', ''y'') is an analytic function of two complex variables with a power series expansion around {{nowrap| ''x'' {{=}} 0 }}, {{nowrap| ''y'' {{=}} 0 }}. This expansion begins as ''r''(''x'', ''y'') = 1 - (''xy''<sup>3</sup> - ''x''<sup>3</sup>''y'')/24 + (3''xy''<sup>4</sup> + ''x''<sup>2</sup>''y''<sup>3</sup> - ''x''<sup>3</sup>''y''<sup>2</sup> - 3''x''<sup>4</sup>''y'')/48 + …, with its first nonconstant term of total degree four, and so when ''x'' and ''y'' are small, ''r''(''x'', ''y'') will be close to 1. The ''n''-th power will still be close to 1, and if there are common factors in the denominators of the exponents, the resulting comma will be less complex (lower in height), and this is the really interesting case. For example, if ''a'' = 7/6 and ''b'' = 27/25, we obtain (7/6)<sup>1/26</sup>/(27/25)<sup>1/13</sup>, the lcm of 13 and 26 is 26, and taking the 26th power gives us 4375/4374. The common factor of 13 leads to the resulting comma (4375/4374) being relatively simple.