us that the lowest power of a power series solution to the Kummer equation is either 0 or . If we let be then the differential equation gives which, upon dividing out and simplifying, becomes This means that is a solution so long as is not an integer greater than 1, just as is a solution so long as is not an integer less than 1. We can also use the Tricomi confluent hypergeometric function introduced by , and sometimes denoted by . It is a combination of the above two solutions, defined by Although this expression is undefined for integer , it has the advantage that it can be extended to any integer by continuity. Unlike Kummer's function which is an entire function of , usually has a singularity at zero. For example, if and then is asymptotic to as goes to zero. But see #Special cases for some examples where it is an entire function (polynomial). Note that the solution to Kummer's equation is the same as the solution , see #Kummer's transformation. 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 : When a = 0 we can alternatively use: When this is the exponential integral . A similar problem occurs when is a negative integer and is an integer less than 1. In this case doesn't exist, and is a multiple of A second solution is then of the form: Other equations Confluent Hypergeometric Functions can be used to solve the Extended Confluent Hypergeometric Equation whose general form is given as: Note that for or when the summation involves just one term, it reduces to the conventional Confluent Hypergeometric Equation. Thus Confluent Hypergeometric Functions can be used to solve "most"