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Paul Masson (talk | contribs) Typo |
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Although this expression is undefined for integer {{mvar|b}}, it has the advantage that it can be extended to any integer {{mvar|b}} by continuity. Unlike Kummer's function which is an [[entire function]] of {{mvar|z}}, {{math|''U''(''z'')}} usually has a [[singularity (mathematics)|singularity]] at zero. For example, if {{math|''b'' {{=}} 0}} and {{math|''a'' ≠ 0}} then {{math|Γ(''a''+1)''U''(''a'', ''b'', ''z'') − 1}} is asymptotic to {{math|''az'' ln ''z''}} as {{mvar|z}} goes to zero. But see [[#Special cases]] for some examples where it is an entire function (polynomial).
Note that the solution {{math|''z''<sup>1−''b''</sup>''
For most combinations of real or complex {{mvar|a}} and {{mvar|b}}, the functions {{math|''M''(''a'', ''b'', ''z'')}} and {{math|''U''(''a'', ''b'', ''z'')}} are independent, and if {{mvar|b}} is a non-positive integer, so {{math|''M''(''a'', ''b'', ''z'')}} doesn't exist, then we may be able to use {{math|''z''<sup>1−''b''</sup>''M''(''a''+1−''b'', 2−''b'', ''z'')}} as a second solution. But if {{mvar|a}} is a non-positive integer and {{mvar|b}} is not a non-positive integer, then {{math|''U''(''z'')}} is a multiple of {{math|''M''(''z'')}}. In that case as well, {{math|''z''<sup>1−''b''</sup>''M''(''a''+1−''b'', 2−''b'', ''z'')}} can be used as a second solution if it exists and is different. But when {{mvar|b}} is an integer greater than 1, this solution doesn't exist, and if {{math|1=''b'' = 1}} then it exists but is a multiple of {{math|''U''(''a'', ''b'', ''z'')}} and of {{math|''M''(''a'', ''b'', ''z'')}} In those cases a second solution exists of the following form and is valid for any real or complex {{mvar|a}} and any positive integer {{mvar|b}} except when {{mvar|a}} is a positive integer less than {{mvar|b}}:
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