Chapter 6
Theory of Analytic Functions
Theory of analytic functions is one of major fields of modern mathematics. Its
application covers broad range of topics of natural science. A complex function
f (z), or a function that takes a complex number z as a variable, has various properties
that often differ from those of functions that take a real number x as a variable. In
particular, the analytic functions hold a paramount position in the complex analysis.
In this chapter we explore various features of the analytic functions accordingly.
From a practical point of view, the theory of analytic functions is very frequently
utilized for the calculation of real definite integrals. For this reason, we describe the
related topics together with tangible examples.
The complex plane (or Gaussian plane) can be dealt with as a topological space
where the metric (or distance function) is defined. Since the complex plane has a
two-dimensional extension, we can readily imagine and investigate its topological
feature. Set theory allows us to make an axiomatic approach along with the topology.
Therefore, we introduce basic notions and building blocks of the set theory and
topology.
6.1 Set and Topology
A complex number z is usually expressed as
z ¼ x þ iy,
ð6:1Þ
where x and y are real numbers and i is an imaginary unit. Graphically, the number
z is indicated as a point in a complex plane where the real axis is drawn as abscissa
and the imaginary axis is depicted as ordinate (see Fig. 6.1). Since the complex plane
has a two-dimensional extension, we can readily imagine the domain of variability of
z and make a graphical object for it on the complex plane. A disk-like diagram that is
enclosed with a closed curve C is frequently dealt with in the theory of analytic
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_6
181
Theory of Analytic Functions
Theory of analytic functions is one of major fields of modern mathematics. Its
application covers broad range of topics of natural science. A complex function
f (z), or a function that takes a complex number z as a variable, has various properties
that often differ from those of functions that take a real number x as a variable. In
particular, the analytic functions hold a paramount position in the complex analysis.
In this chapter we explore various features of the analytic functions accordingly.
From a practical point of view, the theory of analytic functions is very frequently
utilized for the calculation of real definite integrals. For this reason, we describe the
related topics together with tangible examples.
The complex plane (or Gaussian plane) can be dealt with as a topological space
where the metric (or distance function) is defined. Since the complex plane has a
two-dimensional extension, we can readily imagine and investigate its topological
feature. Set theory allows us to make an axiomatic approach along with the topology.
Therefore, we introduce basic notions and building blocks of the set theory and
topology.
6.1 Set and Topology
A complex number z is usually expressed as
z ¼ x þ iy,
ð6:1Þ
where x and y are real numbers and i is an imaginary unit. Graphically, the number
z is indicated as a point in a complex plane where the real axis is drawn as abscissa
and the imaginary axis is depicted as ordinate (see Fig. 6.1). Since the complex plane
has a two-dimensional extension, we can readily imagine the domain of variability of
z and make a graphical object for it on the complex plane. A disk-like diagram that is
enclosed with a closed curve C is frequently dealt with in the theory of analytic
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_6
181