second-order ordinary differential equations whose variable coefficients are all linear functions of , because they can be transformed to the Extended Confluent Hypergeometric Equation. Consider the equation: First we move the regular singular point to by using the substitution of , which converts the equation to: with new values of , and . Next we use the substitution: and multiply the equation by the same factor, obtaining: whose solution is where is a solution to Kummer's equation with Note that the square root may give an imaginary or complex number. If it is zero, another solution must be used, namely where is a confluent hypergeometric limit function satisfying As noted below, even the Bessel equation can be solved using confluent hypergeometric functions. Integral representations If , can be represented as an integral thus is the characteristic function of the beta distribution. For with positive real part can be obtained by the Laplace integral The integral defines a solution in the right half-plane . They can also be represented as Barnes integrals where the contour passes to one side of the poles of and to the other side of the poles of . Asymptotic behavior If a solution to Kummer's equation is asymptotic to a power of as , then the power must be . This is in fact the case for Tricomi's solution . Its asymptotic behavior as can be deduced from the integral representations. If , then making a change of variables in the integral followed by expanding the binomial series and integrating it formally term by term gives rise to an asymptotic series expansion, valid as : where is a generalized hypergeometric series with 1 as leading term, which generally converges nowhere, but exists as a formal power series in . This asymptotic expansion is also valid for complex instead of real , with The asymptotic behavior of Kummer's solution for large is: The powers of are taken using . The first term is not needed when is finite, that is when is not a non-positive integer and the real part of goes to negative infinity, whereas the second term is not needed when is finite, that is, when is a not a non-positive integer and the real part of goes to positive infinity. There is always some solution to Kummer's equation asymptotic to as . Usually this will be a combination of both and but can also