I dont know whether I am the first one who realizes that regularly iterating \( \exp \) at a complex fixed point yields real coefficients! Moreover they do not depend on the chosen fixed point!
To be more precise:
If we have any fixed point \( a \) of \( \exp \), then the power series \( \tau_{-a}\circ \exp\circ \tau_a \) (where \( \tau_a(x):=x+a \)) is a power series with fixed point 0 and with first coefficient \( \exp'(a)=\exp(a)=a \), particularly the series is non-real. By the previous considerations the fixed point is repelling so we have the standard way of hyperbolic iteration (on the main branch) which yields again a non-real power series. If we however afterwards apply the inverse transformation
\( \exp^{\circ t}:=\tau_a\circ (\tau_a^{-1}\circ \exp\circ \tau_a)^{\circ t}\circ \tau_a^{-1} \)
we get back real coefficients and they do not depend on the fixed point \( a \). Though I can not prove it yet, this seems to be quite reliable.
And of course those coefficients are equal to those obtained by the matrix operator method, which in turn equals Andrew's solution after transformation.
To be more precise:
If we have any fixed point \( a \) of \( \exp \), then the power series \( \tau_{-a}\circ \exp\circ \tau_a \) (where \( \tau_a(x):=x+a \)) is a power series with fixed point 0 and with first coefficient \( \exp'(a)=\exp(a)=a \), particularly the series is non-real. By the previous considerations the fixed point is repelling so we have the standard way of hyperbolic iteration (on the main branch) which yields again a non-real power series. If we however afterwards apply the inverse transformation
\( \exp^{\circ t}:=\tau_a\circ (\tau_a^{-1}\circ \exp\circ \tau_a)^{\circ t}\circ \tau_a^{-1} \)
we get back real coefficients and they do not depend on the fixed point \( a \). Though I can not prove it yet, this seems to be quite reliable.
And of course those coefficients are equal to those obtained by the matrix operator method, which in turn equals Andrew's solution after transformation.
