Back to Hans Lundmark's *Complex analysis* page.

Consider the exponential function
*e ^{z}*
and its
Taylor
polynomials

The series for *e ^{z}* converges for all

Well, not really.
A polynomial cannot possibly approximate
*e ^{z}*
in the whole complex plane.
For example,

Recall the domain coloring plot of the exponential function,
here shown on a square with corners at
*z* = ±20±20*i* :

Now look at the plots (on the same square) of the Taylor polynomials
*P*_{5} ,
*P*_{10} ,
*P*_{15} ,
*P*_{20} ,
and
*P*_{25} :

(Side note: Do you get the impression that these images are rotating? Then check out Akiyoshi's illusion pages for a mindboggling collection of similar optical illusions.)

As we anticipated,
the zeros of *P _{n}*
move further and further away from the origin as

The zeros of
*P _{n}*(

**Simple fact**:
The zeros of
*Q _{n}*(

Proof:
The coefficients of
*Q _{n}*
are positive and nondecreasing,
since the coefficient of

(Sketch of proof of the Eneström–Kakeya theorem:
multiply the polynomial by *z*−1
and use the triangle inequality to show that the resulting polynomial
is nonzero for | *z* | < 1.)

**Deep fact**
(proved by
Szegő
in 1924):
As *n* goes to infinity,
the zeros of
*Q _{n}*(

Problems of this type are still an active research topic.
One might study the rate of convergence,
or how densely the zeros accumulate on different parts of the curve.
(See for example the paper
Zeros of the partial sums of cos (*z*) and sin (*z*). I by
Richard S. Varga
and Amos J. Carpenter, Numerical Algorithms **25**: 363-375, 2000.)

Asymptotic behavior of orthogonal polynomials (and their zeros) is a closely related subject where powerful new techniques have been developed in recent years. Details about this can be found in the book Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach by Percy Deift.

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