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:

Analyticity

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:

Periodicity

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 :

Homogeneity

The six complex components , , , , , and have the following homogeneity properties:

Scale symmetry

Some complex components have scale symmetry:

Series representations

The functions and with real have the following series expansions near point :

Integral representations

The function with real has the following contour integral representation:

Limit representations

The functions and with real have the following limit representations:

The last two representations are sometimes called generalized Padé approximations.

Transformations

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:

Complex characteristics

The next two tables describe all the complex components applied to all complex components , , , , , and at the points and :

Differentiation

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.

Indefinite integration

The indefinite integrals of some complex components at the real point can be represented by the following formulas:

Definite integration

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 ):

Integral transforms

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:

Differential equations

The absolute value function for real satisfies the following simple first-order differential equation understandable in a distributional sense:

In a similar manner:

Inequalities

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:

Zeros

The six complex components , , , , , and have the set of zeros described by the following formulas: