Introduction to the complex components
The study of complex numbers and their characteristics has a long history. It all started with questions about how to understand and interpret the solution of the simple quadratic equation .
It was clear that . But it was not clear how to get –1 from something squared.
This problem was intensively discussed in the 16th, 17th, and 18th centuries. As a result, mathematicians proposed a special symbol—the imaginary unit , which is represented by :
L. Euler (1755) introduced the word "complex" (1777) and first used the letter for denoting . Later, C. F. Gauss (1831) introduced the name "imaginary unit" for .
Accordingly, and and the above quadratic equation has two solutions as is expected for a quadratic polynomial:
The imaginary unit was interpreted in a geometrical sense as the point with coordinates in the Cartesian (Euclidean) ,‐plane with the vertical -axis upward and the origin . This geometric interpretation established the following representations of the complex number through two real numbers and as:
where is the distance between points and , and is the angle between the line connecting the points and and the positive -axis direction (the so-called polar representation).
The last formula lead to the basic relations:
which describe the main characteristics of the complex number —the so-called modulus (absolute value) , the real part , the imaginary part , and the argument .
A new era in the theory of complex numbers and functions of complex arguments (analytic functions) arose from the investigations of L. Euler (1727, 1728). In a letter to Goldbach (1731) L. Euler introduced the notation ⅇ for the base of the natural logarithm ⅇ⩵2.71828182…, and he proved that ⅇ is irrational. Later on L. Euler (1740–1748) found a series expansion for , which lead to the famous very basic formula, connecting exponential and trigonometric functions:
This is known as the Euler formula (although it was already derived by R. Cotes in 1714).
The Euler formula allows presentation of the complex number , using polar coordinates in the more compact form:
It also expressed the logarithm of complex numbers through the formula:
Taking into account that the cosine and sine have period , it follows that has period :
Generically, the logarithm function is the multivalued function:
For specifying just one value for the logarithm and one value of the argument φ for a given complex number , the restriction π < φ ≤ π for the argument φ is generally used.
During the 18th and 19th centuries many mathematicians worked on building the theory of the functions of complex variables, which was called the theory of analytic functions. Today this is a widely used theory, not only for the above‐mentioned four complex components (absolute value, argument, real and imaginary parts), but for complimentary characteristics of a complex number such as the conjugate complex number and the signum (sign) . J. R. Argand (1806, 1814) introduced the word "module" for the absolute value, and A. L. Cauchy (1821) was the first to use the word "conjugate" for complex numbers in the modern sense. Later K. Weierstrass (1841) introduced the notation ❘z❘ for the absolute value.
It was shown that the set of complex numbers and the set of real numbers have basic properties in common—they both are fields because they satisfy so-called field axioms. Complex and real numbers exhibit commutativity under addition and multiplication described by the formulas:
Complex and real numbers also have associativity under addition and multiplication described by the formulas:
and distributivity described by the formulas:
(The set of rational numbers also satisfies all of the previous field axioms and is also a field. This set is countable, which means that each rational number can be numerated and placed in a definite position with a corresponding integer number . But the set of rational numbers does not include so-called irrational numbers like or . The set of irrational numbers is much larger and cannot be numerated. The sets of all real and complex numbers form uncountable sets.)
The great success and achievements of the complex number theory stimulated attempts to introduce not only the imaginary unit in the Cartesian (Euclidean) plane , but a similar special third unit in Cartesian (Euclidean) three-dimensional space , which can be used for building a similar theory of (hyper)complex numbers :
Unfortunately, such an attempt fails to fulfill the field axioms. Further generalizations to build the so‐called quaternions and octonions are needed to obtain mathematically interesting and rich objects.
Definitions of complex components
The complex components include six basic characteristics describing complex numbersabsolute value (modulus) , argument (phase) , real part , imaginary part , complex conjugate , and sign function (signum) . It is impossible to define real and imaginary parts of the complex number through other functions or complex characteristics. They are too basic, so their symbols can be described by simple sentences, for example, " gives the real part of the number ," and " gives the imaginary part of the number ."
All other complex components are defined by the following formulas:
Geometrically, the absolute value (or modulus) of a complex number is the Euclidean distance from to the origin, which can also be described by the formula:
Geometrically, the argument of a complex number is the phase angle (in radians) that the line from 0 to makes with the positive real axis. So, the complex number can be presented by the formulas:
Geometrically, the real part of a complex number is the projection of the complex point on the real axis. So, the real part of the complex number can be presented by the formulas:
Geometrically, the imaginary part of a complex number is the projection of complex point on the imaginary axis. So, the imaginary part of the complex number can be presented by the formulas:
Geometrically, the complex conjugate of a complex number is the complex point , which is symmetrical to with respect to the real axis. So, the conjugate value of the complex number can be presented by the formulas:
Geometrically, the sign function (signum) is the complex point that lays on the intersection of the unit circle and the line from 0 to (if ). So, the conjugate value of the complex number can be presented by the formulas:
A quick look at the complex components
Here is a quick look at the graphics for the complex components of the complex components over the complex ‐plane. The empty graphic indicates that the function value is not real.
Connections within the group of complex components and with other function groups
Representations through more general functions
All six complex component functions , , , , , and cannot be easily represented by more generalized functions because most of them are analytic functions of their arguments. But sometimes such representations can be found through Meijer G functions, for example:
Representations through other functions
All six complex components , , , , , and satisfy numerous internal relations of the type , where and are different complex components and is a basic arithmetic operation or (composition of) elementary functions. The most important of these relations are represented in the following table:
Other internal relations between complex components of the type , where and are different complex components that also exist. Some of them are shown here:
Here are some more formulas of the last type:
(here is the Heaviside theta function, also called the unit step function).
The first table can be rewritten using the notations :
The best-known properties and formulas for complex components
Real values for real arguments
For real values of argument , the values of all six complex components , , , , , and are real.
Simple values at zero
The six complex components , , , , , and have the following values for the argument :
is not a uniquely defined number. Depending on the argument of , the limit can take any value in the interval .
Specific values for specialized variable
The six complex components , , , , , and have the following values for some concrete numeric arguments:
Restricted arguments have the following formulas for the six complex components , , , , , and :
The values of complex components , , , , , and at any infinity can be described through the following:
All six complex components , , , , , and are not analytical functions. None of them fulfills the Cauchy–Riemann conditions and as such the value of the derivative depends on the direction. The functions , , , and are real‐analytic functions of the variable (except, maybe, ). The real and the imaginary parts of and are real‐analytic functions of the variable .
Sets of discontinuity
The four complex components , , , and are continuous functions in .
The function has discontinuity at point .
The function is a single‐valued, continuous function on the ‐plane cut along the interval , where it is continuous from above. Its behavior can be described by the following formulas:
All six complex components , , , , , and do not have any periodicity.
Parity and symmetry
All six complex components , , , , , and have mirror symmetry:
The absolute value is an even function. The four complex components , , , and are odd functions. The argument is an odd function for almost all :
The six complex components , , , , , and have the following homogeneity properties:
Some complex components have scale symmetry:
The functions and with real have the following series expansions near point :
The function with real has the following contour integral representation:
The functions and with real have the following limit representations:
The last two representations are sometimes called generalized Padé approximations.
The values of all complex components , , , , , and at the points , , –ⅈ z, , , , , and are given by the following identities:
The values of all complex components , , , , , and at the points , , and are described by the following table:
Some complex components can be easily evaluated in more general cases of the points including symbolic sums and products of , , for example:
The previous tables and formulas can be modified or simplified for particular cases when some variables become real or satisfy special restrictions, for example:
Taking into account that complex components have numerous representations through other complex components and elementary functions such as the logarithm, exponential function, or the inverse tangent function, all of the previous formulas can be transformed into different equivalent forms. Here are some of the resulting formulas for the power function :
Similar identities can be derived for the exponent functions, such as:
Some arithmetical operations involving complex components or elementary functions of complex components are:
The next two tables describe all the complex components applied to all complex components , , , , , and at the points and :
The derivatives of five complex components , , , , and at the real point can be interpreted in a real‐analytic or distributional sense and are given by the following formulas:
where is the Dirac delta function.
It is impossible to make a classical, direction-independent interpretation of these derivatives for complex values of variable because the complex components do not fulfill the Cauchy-Riemann conditions.
The indefinite integrals of some complex components at the real point can be represented by the following formulas:
The definite integrals of some complex components in the complex plane can also be represented through complex components, for example:
Some definite integrals including absolute values can be easily evaluated, for example (in the Hadamard sense of integration, the next identity is correct for all complex values of ):
Fourier integral transforms of the absolute value and signum functions and can be evaluated through generalized functions:
Laplace integral transforms of these functions can be evaluated in a classical sense and have the following values:
The absolute value function for real satisfies the following simple first-order differential equation understandable in a distributional sense:
In a similar manner:
All six complex components , , , , and satisfy numerous inequalities. The best known are so-called triangle inequalities for absolute values:
Some other inequalities can be described by the following formulas:
The six complex components , , , , , and have the set of zeros described by the following formulas:
Applications of complex components
All six complex components are used throughout mathematics, the exact sciences, and engineering.