For most combinations of real or complex and , the functions and are independent, and if is a non-positive integer, so doesn't exist, then we may be able to use as a second solution. But if is a non-positive integer and is not a non-positive integer, then is a multiple of . In that case as well, can be used as a second solution if it exists and is different. But when is an integer greater than 1, this solution doesn't exist, and if then it exists but is a multiple of and of In those cases a second solution exists of the following form and is valid for any real or complex and any positive integer except when is a positive integer less than :