the equation f(x)′−f(x)=0 holds for the exponential function on the complex plane.Now what i dont understand is this."let f ( x ) = a 0 + a 1 X + a 2 X 2 . . . . . . . . f is a polynomial with infinite degree ".Why is that. I dont understand how he came to that conclusion?I mean Why define it that way?.MAybe he could solve the ODE on the real numbers and avoid this "out of nowhere" polynomial or is there a connection?

Question

the equation f(x)′−f(x)=0 holds for the exponential function on the complex plane.Now what i dont understand is this.&quot;let   f  (  x  )  =      a    0    +      a    1    X  +      a    2        X    2    .  .  .  .  .  .  .  . f is a polynomial with infinite degree &quot;.Why is that. I dont understand how he came to that conclusion?I mean Why define it that way?.MAybe he could solve the ODE on the real numbers and avoid this &quot;out of nowhere&quot; polynomial or is there a connection?

Shiloh Davenport · Accepted Answer

The exponential function is holomorphic (complex differentiable, that is) over the complex plane. A fundamental property of holomorphic functions is that they are also analytic over the same domain of definition, therefore, by the definition of an analytic function, you may find a corresponding power series in a neighborood of every point. Hence the polynomial. I&#039;d suggest Stein&#039;s lectures on Complex Analysis for more detail.