In complex analysis, a complex function f(z) of a complex variable z
is called holomorphic or analytic if it is complex differentiable
in the vicinity of every point of its domain.
This is a very strong condition.
As a result, holomorphic functions are infinitely differentiable and
equal their Taylor expansion at every point. In physicists’ terms,
they are very “well-behaved” throughout their domain.
More formally, a given function f(z) is holomorphic in a certain region
if the following limit exists for all z in that region,
and for all directions of Δz:
We decompose f into the real functions u and v of real variables x and y:
Since we are free to choose the direction of Δz, we choose Δx and Δy:
For f(z) to be holomorphic, these two results must be equivalent.
Because u and v are real by definition,
we thus arrive at the Cauchy-Riemann equations:
Therefore, a given function f(z) is holomorphic if and only if
its real and imaginary parts satisfy these equations.
This gives an idea of how strict the criteria are to qualify as holomorphic.
Holomorphic functions satisfy Cauchy’s integral theorem, which states
that the integral of f(z) over any closed curve C in the complex plane is zero,
provided that f(z) is holomorphic for all z in the area enclosed by C:
Just like before, we decompose f(z) into its real and imaginary parts:
Since f(z) is holomorphic, u and v satisfy the Cauchy-Riemann equations,
such that the integrands disappear and the final result is zero.
An interesting consequence is Cauchy’s integral formula,
which states that the value of f(z) at an arbitrary point z0
is determined by its values on an arbitrary contour C around z0:
Thanks to the integral theorem, we know that
the shape and size of C are irrelevant.
Therefore we choose it to be a circle with radius r,
such that the integration variable becomes z=z0+reiθ.
Then we integrate by substitution: