Use the Laplace kernel method to write integral representations in the complex $t$-plane for two linearly independent solutions of the confluent hypergeometric equation

$$z \frac{d^{2} w(z)}{d z^{2}}+(c-z) \frac{d w(z)}{d z}-a w(z)=0$$

in the case that $\operatorname{Re}(z)>0, \operatorname{Re}(c)>\operatorname{Re}(a)>0, a$ and $c-a$ are not integers.