In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
:
where
and
are parameters. They were first introduced by
Émile Léonard Mathieu
Émile Léonard Mathieu (; 15 May 1835, in Metz – 19 October 1890, in Nancy) was a French mathematician. He is known for his work in group theory and mathematical physics. He has given his name to the Mathieu functions, Mathieu groups and Mathi ...
, who encountered them while studying vibrating elliptical drumheads.
[Morse and Feshbach (1953).][Brimacombe, Corless and Zamir (2021)] They have applications in many fields of the physical sciences, such as
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
,
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, and
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. They tend to occur in problems involving periodic motion, or in the analysis of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
s possessing
elliptic
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
symmetry.
[Gutiérrez-Vega (2015).]
Definition
Mathieu functions
In some usages, ''Mathieu function'' refers to solutions of the Mathieu differential equation for arbitrary values of
and
. When no confusion can arise, other authors use the term to refer specifically to
- or
-periodic solutions, which exist only for special values of
and
.
[Arscott (1964), chapter III] More precisely, for given (real)
such periodic solutions exist for an infinite number of values of
, called ''characteristic numbers'', conventionally indexed as two separate sequences
and
, for
. The corresponding functions are denoted
and
, respectively. They are sometimes also referred to as ''cosine-elliptic'' and ''sine-elliptic'', or Mathieu functions of the first kind.
As a result of assuming that
is real, both the characteristic numbers and associated functions are real-valued.
and
can be further classified by
parity and periodicity (both with respect to
), as follows:
:
The indexing with the integer
, besides serving to arrange the characteristic numbers in ascending order, is convenient in that
and
become proportional to
and
as
. With
being an integer, this gives rise to the classification of
and
as Mathieu functions (of the first kind) of integral order. For general
and
, solutions besides these can be defined, including Mathieu functions of fractional order as well as non-periodic solutions.
Modified Mathieu functions
Closely related are the ''modified Mathieu functions'', also known as radial Mathieu functions, which are solutions of ''Mathieu's modified differential equation''
:
which can be related to the original Mathieu equation by taking
. Accordingly, the modified Mathieu functions of the first kind of integral order, denoted by
and
, are defined from
[McLachlan (1947), chapter II.]
:
These functions are real-valued when
is real.
Normalization
A common normalization, which will be adopted throughout this article, is to demand
:
as well as require
and
as
.
Floquet theory
Many properties of the Mathieu differential equation can be deduced from the general theory of ordinary differential equations with periodic coefficients, called
Floquet theory
Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form
:\dot = A(t) x,
with \displaystyle A(t) a Piecewise#Continuity, piecewise contin ...
. The central result is ''Floquet's theorem'':
It is natural to associate the characteristic numbers
with those values of
which result in
. Note, however, that the theorem only guarantees the existence of at least one solution satisfying
, when Mathieu's equation in fact has two independent solutions for any given
,
. Indeed, it turns out that with
equal to one of the characteristic numbers, Mathieu's equation has only one periodic solution (that is, with period
or
), and this solution is one of the
,
. The other solution is nonperiodic, denoted
and
, respectively, and referred to as a Mathieu function of the second kind. This result can be formally stated as ''Ince's theorem'':
An equivalent statement of Floquet's theorem is that Mathieu's equation admits a complex-valued solution of form
:
where
is a complex number, the ''Floquet exponent'' (or sometimes ''Mathieu exponent''), and
is a complex valued function periodic in
with period
. An example
is plotted to the right.
Other types of Mathieu functions
Second kind
Since Mathieu's equation is a second order differential equation, one can construct two linearly independent solutions. Floquet's theory says that if
is equal to a characteristic number, one of these solutions can be taken to be periodic, and the other nonperiodic. The periodic solution is one of the
and
, called a Mathieu function of the first kind of integral order. The nonperiodic one is denoted either
and
, respectively, and is called a Mathieu function of the second kind (of integral order). The nonperiodic solutions are unstable, that is, they diverge as
.
The second solutions corresponding to the modified Mathieu functions
and
are naturally defined as
and
.
Fractional order
Mathieu functions of fractional order can be defined as those solutions
and
,
a non-integer, which turn into
and
as
.
If
is irrational, they are non-periodic; however, they remain bounded as
.
An important property of the solutions
and
, for
non-integer, is that they exist for the same value of
. In contrast, when
is an integer,
and
never occur for the same value of
. (See Ince's Theorem above.)
These classifications are summarized in the table below. The modified Mathieu function counterparts are defined similarly.
:
Explicit representation and computation
First kind
Mathieu functions of the first kind can be represented as
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
:
:
The expansion coefficients
and
are functions of
but independent of
. By substitution into the Mathieu equation, they can be shown to obey three-term
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s in the lower index. For instance, for each
one finds
:
Being a second-order recurrence in the index
, one can always find two independent solutions
and
such that the general solution can be expressed as a linear combination of the two:
. Moreover, in this particular case, an asymptotic analysis
[Wimp (1984), pp. 83-84] shows that one possible choice of fundamental solutions has the property
:
In particular,
is finite whereas
diverges. Writing
, we therefore see that in order for the Fourier series representation of
to converge,
must be chosen such that
These choices of
correspond to the characteristic numbers.
In general, however, the solution of a three-term recurrence with variable coefficients
cannot be represented in a simple manner, and hence there is no simple way to determine
from the condition
. Moreover, even if the approximate value of a characteristic number is known, it cannot be used to obtain the coefficients
by numerically iterating the recurrence towards increasing
. The reason is that as long as
only approximates a characteristic number,
is not identically
and the divergent solution
eventually dominates for large enough
.
To overcome these issues, more sophisticated semi-analytical/numerical approaches are required, for instance using a
continued fraction
In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...
expansion,
casting the recurrence as a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
eigenvalue problem, or implementing a backwards recurrence algorithm.
The complexity of the three-term recurrence relation is one of the reasons there are few simple formulas and identities involving Mathieu functions.
[Temme (2015), p. 234]
In practice, Mathieu functions and the corresponding characteristic numbers can be calculated using pre-packaged software, such as
Mathematica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...
,
Maple
''Acer'' () is a genus of trees and shrubs commonly known as maples. The genus is placed in the family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since http ...
,
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...
, and
SciPy
SciPy (pronounced "sigh pie") is a free and open-source Python library used for scientific computing and technical computing.
SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, signal ...
. For small values of
and low order
, they can also be expressed perturbatively as power series of
, which can be useful in physical applications.
Second kind
There are several ways to represent Mathieu functions of the second kind. One representation is in terms of
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
s:
[Malits (2010)]
:
where
, and
and
are Bessel functions of the first and second kind.
Modified functions
A traditional approach for numerical evaluation of the modified Mathieu functions is through Bessel function product series.
[Jin and Zhang (1996)] For large
and
, the form of the series must be chosen carefully to avoid subtraction errors.
[Van Buren and Boisvert (2007)][Bibby and Peterson (2013)]
Properties
There are relatively few analytic expressions and identities involving Mathieu functions. Moreover, unlike many other special functions, the solutions of Mathieu's equation cannot in general be expressed in terms of
hypergeometric functions. This can be seen by transformation of Mathieu's equation to algebraic form, using the change of variable
:
:
Since this equation has an irregular singular point at infinity, it cannot be transformed into an equation of the hypergeometric type.
Qualitative behavior
![Ce1 vary q](https://upload.wikimedia.org/wikipedia/commons/a/a3/Ce1_vary_q.png)
For small
,
and
behave similarly to
and
. For arbitrary
, they may deviate significantly from their trigonometric counterparts; however, they remain periodic in general. Moreover, for any real
,
and
have exactly
simple zero
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root.
The notion of multipl ...
s in
, and as
the zeros cluster about
.
For
and as
the modified Mathieu functions tend to behave as damped periodic functions.
In the following, the
and
factors from the Fourier expansions for
and
may be referenced (see
Explicit representation and computation
Explicit refers to something that is specific, clear, or detailed. It can also mean:
* Explicit knowledge, knowledge that can be readily articulated, codified and transmitted to others
* Explicit (text) The explicit (from Latin ''explicitus est'', ...
). They depend on
and
but are independent of
.
Reflections and translations
Due to their parity and periodicity,
and
have simple properties under reflections and translations by multiples of
:
:
One can also write functions with negative
in terms of those with positive
:
:
Moreover,
:
Orthogonality and completeness
Like their trigonometric counterparts
and
, the periodic Mathieu functions
and
satisfy orthogonality relations
:
Moreover, with
fixed and
treated as the eigenvalue, the Mathieu equation is of
Sturm–Liouville form. This implies that the eigenfunctions
and
form a complete set, i.e. any
- or
-periodic function of
can be expanded as a series in
and
.
Integral identities
Solutions of Mathieu's equation satisfy a class of integral identities with respect to
kernels
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
that are solutions of
:
More precisely, if
solves Mathieu's equation with given
and
, then the integral
:
where
is a path in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, also solves Mathieu's equation with the same
and
, provided the following conditions are met:
*
solves
*In the regions under consideration,
exists and
is
analytic
*
has the same value at the endpoints of
Using an appropriate change of variables, the equation for
can be transformed into the
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
and solved. For instance, one solution is
. Examples of identities obtained in this way are
:
Identities of the latter type are useful for studying asymptotic properties of the modified Mathieu functions.
There also exist integral relations between functions of the first and second kind, for instance:
:
valid for any complex
and real
.
Asymptotic expansions
The following asymptotic expansions hold for
,
,
, and
:
[McLachlan (1947), chapter XI]
:
Thus, the modified Mathieu functions decay exponentially for large real argument. Similar asymptotic expansions can be written down for
and
; these also decay exponentially for large real argument.
For the even and odd periodic Mathieu functions
and the associated characteristic numbers
one can also derive asymptotic expansions for large
. For the characteristic numbers in particular, one has with
approximately an odd integer, i.e.
:
Observe the symmetry here in replacing
and
by
and
, which is a significant feature of the expansion. Terms of this expansion have been obtained explicitly up to and including the term of order
.
[Dingle and Müller (1962)] Here
is only approximately an odd integer because in the limit of
all minimum segments of the periodic potential
become effectively independent harmonic oscillators (hence
an odd integer). By decreasing
, tunneling through the barriers becomes possible (in physical language), leading to a splitting of the characteristic numbers
(in quantum mechanics called eigenvalues) corresponding to even and odd periodic Mathieu functions. This splitting is obtained with boundary conditions
(in quantum mechanics this provides the splitting of the eigenvalues into energy bands).
[Müller-Kirsten (2012)] The boundary conditions are:
:
Imposing these boundary conditions on the asymptotic periodic Mathieu functions associated with the above expansion for
one obtains
:
The corresponding characteristic numbers or eigenvalues then follow by expansion, i.e.
:
Insertion of the appropriate expressions above yields the result
:
For
these are the eigenvalues associated with the even Mathieu eigenfunctions
or
(i.e. with upper, minus sign) and odd Mathieu eigenfunctions
or
(i.e. with lower, plus sign). The explicit and normalised expansions of the eigenfunctions can be found in
or.
Similar asymptotic expansions can be obtained for the solutions of other periodic differential equations, as for
Lamé function
In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation. It was introduced in the paper . Lamé's equation appears in the method of separation of variable ...
s and prolate and oblate
spheroidal wave function
Spheroidal wave functions are solutions of the Helmholtz equation that are found by writing the equation in spheroidal coordinates and applying the technique of separation of variables, just like the use of spherical coordinates lead to spherical h ...
s.
Applications
Mathieu's differential equations appear in a wide range of contexts in engineering, physics, and applied mathematics. Many of these applications fall into one of two general categories: 1) the analysis of partial differential equations in elliptic geometries, and 2) dynamical problems which involve forces that are periodic in either space or time. Examples within both categories are discussed below.
Partial differential equations
Mathieu functions arise when
separation of variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
in elliptic coordinates is applied to 1) the
Laplace equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \nab ...
in 3 dimensions, and 2) the
Helmholtz equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation
\nabla^2 f = -k^2 f,
where is the Laplace operator (or "Laplacian"), is the eigenv ...
in either 2 or 3 dimensions. Since the Helmholtz equation is a prototypical equation for modeling the spatial variation of classical waves, Mathieu functions can be used to describe a variety of wave phenomena. For instance, in
computational electromagnetics
Computational electromagnetics (CEM), computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment.
It typically involves using computer ...
they can be used to analyze the
scattering
Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...
of
electromagnetic wave
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
s off elliptic cylinders, and wave propagation in elliptic
waveguide
A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
s. In
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, an exact plane wave solution to the
Einstein field equation
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
can be given in terms of Mathieu functions.
More recently, Mathieu functions have been used to solve a special case of the
Smoluchowski equation
Marian Smoluchowski (; 28 May 1872 – 5 September 1917) was a Polish physicist who worked in the Polish territories of the Austro-Hungarian Empire. He was a pioneer of statistical physics, and an avid mountaineer.
Life
Born into an upper- ...
, describing the steady-state statistics of
self-propelled particles.
The remainder of this section details the analysis for the two-dimensional Helmholtz equation. In rectangular coordinates, the Helmholtz equation is
:
Elliptic coordinates
In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci F_ and F_ are generally taken to be fixed at -a and +a, respectively ...
are defined by
:
where
,
, and
is a positive constant. The Helmholtz equation in these coordinates is
:
The constant
curves are
confocal ellipses with focal length
; hence, these coordinates are convenient for solving the Helmholtz equation on domains with elliptic boundaries. Separation of variables via
yields the Mathieu equations
:
where
is a separation constant.
As a specific physical example, the Helmholtz equation can be interpreted as describing
normal mode
A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. ...
s of an elastic membrane under uniform
tension
Tension may refer to:
Science
* Psychological stress
* Tension (physics), a force related to the stretching of an object (the opposite of compression)
* Tension (geology), a stress which stretches rocks in two opposite directions
* Voltage or el ...
. In this case, the following physical conditions are imposed:
*Periodicity with respect to
, i.e.
*Continuity of displacement across the interfocal line:
*Continuity of derivative across the interfocal line:
For given
, this restricts the solutions to those of the form
and
, where
. This is the same as restricting allowable values of
, for given
. Restrictions on
then arise due to imposition of physical conditions on some bounding surface, such as an elliptic boundary defined by
. For instance, clamping the membrane at
imposes
, which in turn requires
:
These conditions define the normal modes of the system.
Dynamical problems
In dynamical problems with periodically varying forces, the
equation of motion
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Verla ...
sometimes takes the form of Mathieu's equation. In such cases, knowledge of the general properties of Mathieu's equation— particularly with regard to stability of the solutions—can be essential for understanding qualitative features of the physical dynamics.
[Meixner and Schäfke (1954), pp. 324–343] A classic example along these lines is the
inverted pendulum
An inverted pendulum is a pendulum that has its center of mass above its pivot point. It is unstable and without additional help will fall over. It can be suspended stably in this inverted position by using a control system to monitor the angle ...
. Other examples are
*vibrations of a string with periodically varying tension
*stability of railroad rails as trains drive over them
*seasonally forced
population dynamics
Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems.
History
Population dynamics has traditionally been the dominant branch of mathematical biology, which has ...
*the phenomenon of
parametric resonance in forced
oscillator
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
s
*motion of ions in a
quadrupole ion trap
A quadrupole ion trap or paul trap is a type of ion trap that uses dynamic electric fields to trap charged particles. They are also called radio frequency (RF) traps or Paul traps in honor of Wolfgang Paul, who invented the device and shared the N ...
*the
Stark effect
The Stark effect is the shifting and splitting of spectral lines of atoms and molecules due to the presence of an external electric field. It is the electric-field analogue of the Zeeman effect, where a spectral line is split into several compon ...
for a rotating
electric dipole
The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI unit for electric dipole moment is the coulomb-meter (C⋅m). The d ...
* the
Floquet theory
Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form
:\dot = A(t) x,
with \displaystyle A(t) a Piecewise#Continuity, piecewise contin ...
of the stability of
limit cycles
Quantum mechanics
Mathieu functions play a role in certain quantum mechanical systems, particularly those with spatially periodic potentials such as the
quantum pendulum
The quantum pendulum is fundamental in understanding hindered internal rotations in chemistry, quantum features of scattering atoms, as well as numerous other quantum phenomena. Though a pendulum not subject to the small-angle approximation has an ...
and
crystalline lattice
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
s.
The modified Mathieu equation also arises when describing the quantum mechanics of singular potentials. For the particular singular potential
the radial
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
:
can be converted into the equation
:
The transformation is achieved with the following substitutions
:
By solving the Schrödinger equation (for this particular potential) in terms of solutions of the modified Mathieu equation, scattering properties such as the
S-matrix
In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
More forma ...
and the
absorptivity can be obtained.
[Müller-Kirsten (2006)]
See also
*
List of mathematical functions
In mathematics, some functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. There is a large theory of special functions which developed ...
*
Hill differential equation
In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation
: \frac + f(t) y = 0,
where f(t) is a periodic function by minimal period \pi . By these we mean that for all t
:f(t+\ ...
*
Lamé function
In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation. It was introduced in the paper . Lamé's equation appears in the method of separation of variable ...
*
Monochromatic electromagnetic plane wave In general relativity, the monochromatic electromagnetic plane wave spacetime is the analog of the monochromatic plane waves known from Maxwell's theory. The precise definition of the solution is quite complicated but very instructive.
Any exact ...
*
Inverted pendulum
An inverted pendulum is a pendulum that has its center of mass above its pivot point. It is unstable and without additional help will fall over. It can be suspended stably in this inverted position by using a control system to monitor the angle ...
*
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
Notes
References
*
*
*
*
*
*
*
*
*
*
*
*
*
* ''Note: Reprinted lithographically in Great Britain at the University Press, Oxford, 1951 from corrected sheets of the (1947) first edition.''
*
*
*
*
*
*
*
*
*
* (free online access to the appendix on Mathieu functions)
*
*
*
External links
*
List of equations and identities for Mathieu Functionsfunctions.wolfram.com
*
* Timothy Jones,
' (2006)
*
Mathieu equation'
EqWorldNIST Digital Library of Mathematical Functions: Mathieu Functions and Hill's Equation
{{DEFAULTSORT:Mathieu Function
Ordinary differential equations
Special functions