 an entire function, with MacLaurin series

mimicking the polynomial case suggests we define

Since the MacLaurin series converges everywhere, the above series will converge, in a chosen operator norm.An example of this is the exponential of a matrix.Replacing "z" by "T" in the MacLaurin series of "f"("z") = "ez" gives

The requirement that the MacLaurin series of "f" converges everywhere can be relaxed somewhat.