How to Read the Bessel Function Graph

March 04, 2022 Post a Comment

Introduction to the Bessel functions

General

The Bessel functions take been known since the 18th century when mathematicians and scientists started to depict concrete processes through differential equations. Many different‐looking processes satisfy the same partial differential equations. These equations were named Laplace, d`Alembert (wave), Poisson, Helmholtz, and rut (improvidence) equations. Different methods were used to investigate these equations. The most powerful was the separation of variables method, which in polar coordinates often leads to ordinary differential equations of special structure:

This equation with concrete values of the parameter appeared in the articles past F. W. Bessel (1816, 1824) who built 2 partial solutions and of the previous equation in the form of series:

Substituting the series into the differential equation produces the following solutions:

O. Schlömilch (1857) used the name Bessel functions for these solutions, Eastward. Lommel (1868) considered as an arbitrary real parameter, and H. Hankel (1869) considered complex values for . The two independent solutions of the differential equation were notated every bit and .

For integer alphabetize , the functions and coincide or take different signs. In such cases, the 2d linear independent solution of the previous differential equation was introduced by C. G. Neumann (1867) as the limit case of the following special linear combination of the functions and :

J. Watson (1867) introduced the annotation for this function. Other authors (H. Hankel (1869), H. Weber (1873), and 50. Schläfli (1875)) investigated its properties. In item, the full general solution of the previous differential equation for all values of the parameter can be presented by the formula:

where and are arbitrary circuitous constants.

In a like mode, A. B. Basset (1888) and H. M. MacDonald (1899) introduced the modified Bessel functions and , which satisfy the modified Bessel differential equation:

The start differential equation can be converted into the last i by irresolute the independent variable to .

Definitions of Bessel functions

The Bessel functions of the first kind and are defined as sums of the following infinite series:

These sums are convergent everywhere in the complex ‐aeroplane. The Bessel functions of the 2d kind and for noninteger parameter are defined as special linear combinations of the final ii functions:

In the case of integer index , the correct‐mitt sides of the previous expressions give removable indeterminate values of the type . In this case, the Bessel functions and are defined through the following limits:

A quick look at the Bessel functions

Hither is a quick look at the graphics for the Bessel functions along the existent axis.

Connections within the grouping of Bessel functions and with other function groups

Representations through more full general functions

The Bessel functions , , , and are particular cases of more full general functions: hypergeometric and Meijer G functions.

In particular, the functions and can exist represented through the regularized hypergeometric functions (without whatsoever restrictions on the parameter ):

Similar formulas, merely with restrictions on the parameter , represent and through the classical hypergeometric function :

The functions and tin can too be represented through the hypergeometric functions past the following formulas:

Similar formulas for other Bessel functions and always include restrictions on the parameter, namely :

In the case of integer , the right‐mitt sides of the preceding six formulas evaluate to removable indeterminate expressions of the type , . The limit of the right‐manus sides exists and produces complicated series expansions including logarithmic and polygamma functions. These difficulties can be removed by using the generalized Meijer 1000 role. The generalized Meijer Thousand function allows represention of all four Bessel functions for all values of the parameter past the post-obit elementary formulas:

The classical Meijer Thousand function is less user-friendly because it can lead to additional restrictions:

Representations through other Bessel functions

Each of the Bessel functions tin can exist represented through other Bessel functions:

The best-known properties and formulas for Bessel functions

Real values for real arguments

For real values of parameter and positive argument , the values of all iv Bessel functions , , , and are real.

Simple values at null

The Bessel functions , , , and have rather simple values for the statement :

Specific values for specialized parameters

In the case of half‐integer (ν= ) all Bessel functions , , and can be expressed through sine, cosine, or exponential functions multiplied by rational and square root functions. Modulo simple factors, these are the then‐called spherical Bessel functions, for instance:

The previous formulas are particular cases of the following, more general formulas:

It tin can be shown that for other values of the parameters , the Bessel functions cannot exist represented through elementary functions. Simply for values equal to , and , all Bessel functions tin can exist converted into other known special functions, the Blusterous functions and their derivatives, for instance:

Analyticity

All 4 Bessel functions , , , and are divers for all complex values of the parameter and variable , and they are belittling functions of and over the whole circuitous ‐ and ‐planes.

Poles and essential singularities

For fixed , the functions , , , and have an essential singularity at . At the same time, the betoken is a branch point (except in the example of integer for the two functions ). For fixed integer , the functions and are entire functions of .

For fixed , the functions , , , and are entire functions of and accept merely one essential atypical indicate at .

Branch points and branch cuts

For fixed noninteger , the functions , , , and take two branch points: , , and 1 directly line co-operative cut between them. For fixed integer , just the functions and have two branch points: , , and 1 direct line branch cutting between them.

For cases where the functions , , , and have branch cuts, the branch cuts are single‐valued functions on the ‐plane cut along the interval , where they are continuous from above:

These functions have discontinuities that are described by the following formulas:

Periodicity

All Bessel functions , , , and do not have periodicity.

Parity and symmetry

All Bessel functions , , , and have mirror symmetry (ignoring the interval (-∞, 0)):

The two Bessel functions of the start kind have special parity (either odd or even) in each variable:

The two Bessel functions of the 2nd kind have special parity (either odd or even) only in their parameter:

Serial representations

The Bessel functions , , , and have the post-obit series expansions (which converge in the whole complex ‐plane):

The last four formulas accept restrictions that do not allow their right sides to become indeterminate expressions for integer . In such cases, evaluation of the limit from the right sides leads to much more complicated representations, for example:

Interestingly, airtight‐form expressions for the truncated version of the Taylor series at the origin can be expressed through the generalized hypergeometric role and the Meijer G function, for example:

Asymptotic series expansions

The asymptotic behavior of the Bessel functions , , , and can be described by the following formulas (which prove only the principal terms):

The previous formulas are valid for any direction budgeted the point to infinity (z∞). In item cases, when or , the 2d and fourth formulas can be simplified to the following forms:

Integral representations

The Bessel functions , , , and have uncomplicated integral representations through the cosine (or the hyperbolic cosine or exponential part) and ability functions in the integrand:

Transformations

The argument of the Bessel functions , , , and sometimes tin exist simplified through formulas that remove square roots from the arguments. For the Bessel functions of the 2d kind and with integer index , this performance is realized by special formulas that include logarithms:

If the argument of a Bessel role includes an explicit minus sign, the following formulas produce Bessel functions without the minus sign argument:

If the arguments of the Bessel functions include sums, the following formulas hold:

If arguments of the Bessel functions include products, the post-obit formulas hold:

Identities

The Bessel functions , , , and satisfy the following recurrence identities:

The last eight identities tin can be generalized to the post-obit recurrence identities with jump length :

Unproblematic representations of derivatives

The derivatives of all the four Bessel functions , , , and take rather simple and symmetrical representations that can be expressed through other Bessel functions with unlike indices:

But these derivatives can be represented in other forms, for example:

The symbolic -order derivatives have more than complicated representations through the regularized hypergeometric office or generalized Meijer G function:

Differential equations

The Bessel functions , , , and appeared every bit special solutions of two linear second-order differential equations (the and then‐chosen Bessel equation):

where and are arbitrary constants.

Zeros

When is existent, the functions and each take an space number of existent zeros, all of which are simple with the possible exception of the zip :

When , the zeros of are all real. If and is not an integer, the number of circuitous zeros of is ; if is odd, ii of these zeros lie on the imaginary axis.