Differential equations in the complex plane are different from differential equations on the real line.

Suppose you have an *n*th order linear differential equation as follows.

## The real case

If the *a*‘s are continuous, real-valued functions on an open interval of the real line, then there are *n* linearly independent solutions over that interval. The coefficients do not need to be differentiable for the solutions to be differentiable.

## The complex case

If the *a*‘s are continuous, complex-valued functions on an connected open of the real line, then there may not be *n* linearly independent solutions over that interval.

If the *a*‘s are all analytic, then there are indeed *n* independent solutions. But if any one of the *a*‘s is merely continuous and not analytic, there will be fewer than *n* independent solutions. There may be as many as *n* − 1 solutions, or there may not be any solutions at all.

Source: A. K. Bose. Linear Differential Equations on the Complex Plane. The American Mathematical Monthly, Vol. 89, No. 4 (Apr., 1982), pp 244–246.