Calculus of variations

The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.[a] Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.

A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, where the optical length depends upon the material of the medium. One corresponding concept in mechanics is the principle of least/stationary action.

Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet principle. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in a solution of soap suds. Although such experiments are relatively easy to perform, their mathematical interpretation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.

HistoryEdit

The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696).[2] It immediately occupied the attention of Jakob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733. Lagrange was influenced by Euler's work to contribute significantly to the theory. After Euler saw the 1755 work of the 19-year-old Lagrange, Euler dropped his own partly geometric approach in favor of Lagrange's purely analytic approach and renamed the subject the calculus of variations in his 1756 lecture Elementa Calculi Variationum.[3][4][1]

Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. Isaac Newton and Gottfried Leibniz also gave some early attention to the subject.[5] To this discrimination Vincenzo Brunacci (1810), Carl Friedrich Gauss (1829), Siméon Poisson (1831), Mikhail Ostrogradsky (1834), and Carl Jacobi (1837) have been among the contributors. An important general work is that of Sarrus (1842) which was condensed and improved by Cauchy (1844). Other valuable treatises and memoirs have been written by Strauch (1849), Jellett (1850), Otto Hesse (1857), Alfred Clebsch (1858), and Carll (1885), but perhaps the most important work of the century is that of Weierstrass. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation. The 20th and the 23rd Hilbert problem published in 1900 encouraged further development.[5]

In the 20th century David Hilbert, Emmy Noether, Leonida Tonelli, Henri Lebesgue and Jacques Hadamard among others made significant contributions.[5] Marston Morse applied calculus of variations in what is now called Morse theory.[6] Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory.[6] The dynamic programming of Richard Bellman is an alternative to the calculus of variations.[7][8][9][b]

ExtremaEdit

The calculus of variations is concerned with the maxima or minima (collectively called extrema) of functionals. A functional maps functions to scalars, so functionals have been described as "functions of functions." Functionals have extrema with respect to the elements   of a given function space defined over a given domain. A functional   is said to have an extremum at the function   if   has the same sign for all   in an arbitrarily small neighborhood of  [c] The function   is called an extremal function or extremal.[d] The extremum   is called a local maximum if   everywhere in an arbitrarily small neighborhood of   and a local minimum if   there. For a function space of continuous functions, extrema of corresponding functionals are called strong extrema or weak extrema, depending on whether the first derivatives of the continuous functions are respectively all continuous or not.[11]

Both strong and weak extrema of functionals are for a space of continuous functions but strong extrema have the additional requirement that the first derivatives of the functions in the space be continuous. Thus a strong extremum is also a weak extremum, but the converse may not hold. Finding strong extrema is more difficult than finding weak extrema.[12] An example of a necessary condition that is used for finding weak extrema is the Euler–Lagrange equation.[13][e]

Euler–Lagrange equationEdit

Finding the extrema of functionals is similar to finding the maxima and minima of functions. The maxima and minima of a function may be located by finding the points where its derivative vanishes (i.e., is equal to zero). The extrema of functionals may be obtained by finding functions where the functional derivative is equal to zero. This leads to solving the associated Euler–Lagrange equation.[f]

Consider the functional

 

where

  are constants,
  is twice continuously differentiable,
 
  is twice continuously differentiable with respect to its arguments   and  

If the functional   attains a local minimum at   and   is an arbitrary function that has at least one derivative and vanishes at the endpoints   and   then for any number   close to 0,

 

The term   is called the variation of the function   and is denoted by  [1][g]

Substituting   for   in the functional   the result is a function of  

 

Since the functional   has a minimum for   the function   has a minimum at   and thus,[h]

 

Taking the total derivative of   where   and   are considered as functions of   rather than   yields

 

and because     and   

 

Therefore,

 

where   when   and we have used integration by parts on the second term. The second term on the second line vanishes because   at   and   by definition. Also, as previously mentioned the left side of the equation is zero so that

 

According to the fundamental lemma of calculus of variations, the part of the integrand in parentheses is zero, i.e.

 

which is called the Euler–Lagrange equation. The left hand side of this equation is called the functional derivative of   and is denoted  

In general this gives a second-order ordinary differential equation which can be solved to obtain the extremal function   The Euler–Lagrange equation is a necessary, but not sufficient, condition for an extremum   A sufficient condition for a minimum is given in the section Variations and sufficient condition for a minimum.

ExampleEdit

In order to illustrate this process, consider the problem of finding the extremal function   which is the shortest curve that connects two points   and   The arc length of the curve is given by

 

with

 

[i]

The Euler–Lagrange equation will now be used to find the extremal function   that minimizes the functional  

 

with

 

Since   does not appear explicitly in   the first term in the Euler–Lagrange equation vanishes for all   and thus,

 

Substituting for   and taking the derivative,

 

Thus

 

for some constant   Then

 

where

 

Solving, we get

 

which implies that

 

is a constant and therefore that the shortest curve that connects two points   and   is

 

and we have thus found the extremal function   that minimizes the functional   so that   is a minimum. The equation for a straight line is   In other words, the shortest distance between two points is a straight line.[j]

Beltrami's identityEdit

In physics problems it may be the case that   meaning the integrand is a function of   and   but   does not appear separately. In that case, the Euler–Lagrange equation can be simplified to the Beltrami identity[16]

 

where   is a constant. The left hand side is the Legendre transformation of   with respect to  

The intuition behind this result is that, if the variable   is actually time, then the statement   implies that the Lagrangian is time-independent. By Noether's theorem, there is an associated conserved quantity. In this case, this quantity is the Hamiltonian, the Legendre transform of the Lagrangian, which (often) coincides with the energy of the system. This is (minus) the constant in Beltrami's identity.

Euler–Poisson equationEdit

If   depends on higher-derivatives of   that is, if

 

then   must satisfy the Euler–Poisson equation,

 [17]

Du Bois-Reymond's theoremEdit

The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integral   requires only first derivatives of trial functions. The condition that the first variation vanishes at an extremal may be regarded as a weak form of the Euler–Lagrange equation. The theorem of Du Bois-Reymond asserts that this weak form implies the strong form. If   has continuous first and second derivatives with respect to all of its arguments, and if

 

then   has two continuous derivatives, and it satisfies the Euler–Lagrange equation.

Lavrentiev phenomenonEdit

Hilbert was the first to give good conditions for the Euler–Lagrange equations to give a stationary solution. Within a convex area and a positive thrice differentiable Lagrangian the solutions are composed of a countable collection of sections that either go along the boundary or satisfy the Euler–Lagrange equations in the interior.

However Lavrentiev in 1926 showed that there are circumstances where there is no optimum solution but one can be approached arbitrarily closely by increasing numbers of sections. The Lavrentiev Phenomenon identifies a difference in the infimum of a minimization problem across different classes of admissible functions. For instance the following problem, presented by Manià in 1934:[18]

 
 

Clearly,  minimizes the functional, but we find any function   gives a value bounded away from the infimum.

Examples (in one-dimension) are traditionally manifested across   and   but Ball and Mizel[19] procured the first functional that displayed Lavrentiev's Phenomenon across   and   for   There are several results that gives criteria under which the phenomenon does not occur - for instance 'standard growth', a Lagrangian with no dependence on the second variable, or an approximating sequence satisfying Cesari's Condition (D) - but results are often particular, and applicable to a small class of functionals.

Connected with the Lavrentiev Phenomenon is the repulsion property: any functional displaying Lavrentiev's Phenomenon will display the weak repulsion property.[20]

Functions of several variablesEdit

For example, if   denotes the displacement of a membrane above the domain   in the   plane, then its potential energy is proportional to its surface area:

 

Plateau's problem consists of finding a function that minimizes the surface area while assuming prescribed values on the boundary of  ; the solutions are called minimal surfaces. The Euler–Lagrange equation for this problem is nonlinear:

 

See Courant (1950) for details.

Dirichlet's principleEdit

It is often sufficient to consider only small displacements of the membrane, whose energy difference from no displacement is approximated by

 

The functional   is to be minimized among all trial functions   that assume prescribed values on the boundary of   If   is the minimizing function and   is an arbitrary smooth function that vanishes on the boundary of   then the first variation of   must vanish:

 

Provided that u has two derivatives, we may apply the divergence theorem to obtain

 

where   is the boundary of     is arclength along   and   is the normal derivative of   on   Since   vanishes on   and the first variation vanishes, the result is

 

for all smooth functions v that vanish on the boundary of   The proof for the case of one dimensional integrals may be adapted to this case to show that

 in  

The difficulty with this reasoning is the assumption that the minimizing function u must have two derivatives. Riemann argued that the existence of a smooth minimizing function was assured by the connection with the physical problem: membranes do indeed assume configurations with minimal potential energy. Riemann named this idea the Dirichlet principle in honor of his teacher Peter Gustav Lejeune Dirichlet. However Weierstrass gave an example of a variational problem with no solution: minimize

 

among all functions   that satisfy   and     can be made arbitrarily small by choosing piecewise linear functions that make a transition between −1 and 1 in a small neighborhood of the origin. However, there is no function that makes  [k] Eventually it was shown that Dirichlet's principle is valid, but it requires a sophisticated application of the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998).

Generalization to other boundary value problemsEdit

A more general expression for the potential energy of a membrane is

 

This corresponds to an external force density   in   an external force   on the boundary   and elastic forces with modulus  acting on   The function that minimizes the potential energy with no restriction on its boundary values will be denoted by   Provided that   and   are continuous, regularity theory implies that the minimizing function   will have two derivatives. In taking the first variation, no boundary condition need be imposed on the increment  The first variation of   is given by

 

If we apply the divergence theorem, the result is

 

If we first set   on   the boundary integral vanishes, and we conclude as before that

 

in   Then if we allow   to assume arbitrary boundary values, this implies that   must satisfy the boundary condition

 

on   This boundary condition is a consequence of the minimizing property of  : it is not imposed beforehand. Such conditions are called natural boundary conditions.

The preceding reasoning is not valid if   vanishes identically on   In such a case, we could allow a trial function   where   is a constant. For such a trial function,

 

By appropriate choice of     can assume any value unless the quantity inside the brackets vanishes. Therefore, the variational problem is meaningless unless

 

This condition implies that net external forces on the system are in equilibrium. If these forces are in equilibrium, then the variational problem has a solution, but it is not unique, since an arbitrary constant may be added. Further details and examples are in Courant and Hilbert (1953).

Eigenvalue problemsEdit

Both one-dimensional and multi-dimensional eigenvalue problems can be formulated as variational problems.

Sturm–Liouville problemsEdit

The Sturm–Liouville eigenvalue problem involves a general quadratic form

 

where  is restricted to functions that satisfy the boundary conditions

 

Let   be a normalization integral

 

The functions   and   are required to be everywhere positive and bounded away from zero. The primary variational problem is to minimize the ratio   among all   satisfying the endpoint conditions. It is shown below that the Euler–Lagrange equation for the minimizing   is

 

where   is the quotient

 

It can be shown (see Gelfand and Fomin 1963) that the minimizing   has two derivatives and satisfies the Euler–Lagrange equation. The associated   will be denoted by  ; it is the lowest eigenvalue for this equation and boundary conditions. The associated minimizing function will be denoted by   This variational characterization of eigenvalues leads to the Rayleigh–Ritz method: choose an approximating   as a linear combination of basis functions (for example trigonometric functions) and carry out a finite-dimensional minimization among such linear combinations. This method is often surprisingly accurate.

The next smallest eigenvalue and eigenfunction can be obtained by minimizing   under the additional constraint

 

This procedure can be extended to obtain the complete sequence of eigenvalues and eigenfunctions for the problem.

The variational problem also applies to more general boundary conditions. Instead of requiring that   vanish at the endpoints, we may not impose any condition at the endpoints, and set

 

where   and   are arbitrary. If we set  the first variation for the ratio   is

 

where λ is given by the ratio   as previously. After integration by parts,

 

If we first require that   vanish at the endpoints, the first variation will vanish for all such   only if

 

If   satisfies this condition, then the first variation will vanish for arbitrary   only if

 

These latter conditions are the natural boundary conditions for this problem, since they are not imposed on trial functions for the minimization, but are instead a consequence of the minimization.

Eigenvalue problems in several dimensionsEdit

Eigenvalue problems in higher dimensions are defined in analogy with the one-dimensional case. For example, given a domain   with boundary   in three dimensions we may define

 

and

 

Let   be the function that minimizes the quotient   with no condition prescribed on the boundary   The Euler–Lagrange equation satisfied by   is

 

where

 

The minimizing   must also satisfy the natural boundary condition

 

on the boundary   This result depends upon the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998) for details. Many extensions, including completeness results, asymptotic properties of the eigenvalues and results concerning the nodes of the eigenfunctions are in Courant and Hilbert (1953).

ApplicationsEdit

OpticsEdit

Fermat's principle states that light takes a path that (locally) minimizes the optical length between its endpoints. If the  -coordinate is chosen as the parameter along the path, and   along the path, then the optical length is given by

 

where the refractive index   depends upon the material. If we try   then the first variation of   (the derivative of   with respect to ε) is

 

After integration by parts of the first term within brackets, we obtain the Euler–Lagrange equation

 

The light rays may be determined by integrating this equation. This formalism is used in the context of Lagrangian optics and Hamiltonian optics.

Snell's lawEdit

There is a discontinuity of the refractive index when light enters or leaves a lens. Let

 
 

where   and   are constants. Then the Euler–Lagrange equation holds as before in the region where   or   and in fact the path is a straight line there, since the refractive index is constant. At the     must be continuous, but   may be discontinuous. After integration by parts in the separate regions and using the Euler–Lagrange equations, the first variation takes the form

 

The factor multiplying   is the sine of angle of the incident ray with the   axis, and the factor multiplying   is the sine of angle of the refracted ray with the   axis. Snell's law for refraction requires that these terms be equal. As this calculation demonstrates, Snell's law is equivalent to vanishing of the first variation of the optical path length.

Fermat's principle in three dimensionsEdit

It is expedient to use vector notation: let   let   be a parameter, let   be the parametric representation of a curve   and let   be its tangent vector. The optical length of the curve is given by

 

Note that this integral is invariant with respect to changes in the parametric representation of   The Euler–Lagrange equations for a minimizing curve have the symmetric form

 

where

 

It follows from the definition that   satisfies

 

Therefore, the integral may also be written as

 

This form suggests that if we can find a function   whose gradient is given by   then the integral   is given by the difference of   at the endpoints of the interval of integration. Thus the problem of studying the curves that make the integral stationary can be related to the study of the level surfaces of  In order to find such a function, we turn to the wave equation, which governs the propagation of light. This formalism is used in the context of Lagrangian optics and Hamiltonian optics.

Connection with the wave equationEdit

The wave equation for an inhomogeneous medium is

 

where   is the velocity, which generally depends upon   Wave fronts for light are characteristic surfaces for this partial differential equation: they satisfy

 

We may look for solutions in the form

 

In that case,   satisfies

 

where   According to the theory of first-order partial differential equations, if   then   satisfies

 

along a system of curves (the light rays) that are given by

 

These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification

 

We conclude that the function   is the value of the minimizing integral   as a function of the upper end point. That is, when a family of minimizing curves is constructed, the values of the optical length satisfy the characteristic equation corresponding the wave equation. Hence, solving the associated partial differential equation of first order is equivalent to finding families of solutions of the variational problem. This is the essential content of the Hamilton–Jacobi theory, which applies to more general variational problems.

MechanicsEdit

In classical mechanics, the action,   is defined as the time integral of the Lagrangian,   The Lagrangian is the difference of energies,

 

where   is the kinetic energy of a mechanical system and   its potential energy. Hamilton's principle (or the action principle) states that the motion of a conservative holonomic (integrable constraints) mechanical system is such that the action integral

 

is stationary with respect to variations in the path   The Euler–Lagrange equations for this system are known as Lagrange's equations:

 

and they are equivalent to Newton's equations of motion (for such systems).

The conjugate momenta   are defined by

 

For example, if

 

then

 

Hamiltonian mechanics results if the conjugate momenta are introduced in place of   by a Legendre transformation of the Lagrangian   into the Hamiltonian   defined by

 

The Hamiltonian is the total energy of the system:   Analogy with Fermat's principle suggests that solutions of Lagrange's equations (the particle trajectories) may be described in terms of level surfaces of some function of   This function is a solution of the Hamilton–Jacobi equation:

 

Further applicationsEdit

Further applications of the calculus of variations include the following:

Variations and sufficient condition for a minimumEdit

Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation[l] is defined as the linear part of the change in the functional, and the second variation[m] is defined as the quadratic part.[22]

For example, if   is a functional with the function   as its argument, and there is a small change in its argument from   to   where   is a function in the same function space as   then the corresponding change in the functional is

   [n]

The functional   is said to be differentiable if

 

where   is a linear functional,[o]   is the norm of  [p] and   as   The linear functional   is the first variation of   and is denoted by,[26]

 

The functional   is said to be twice differentiable if

 

where   is a linear functional (the first variation),   is a quadratic functional,[q] and   as   The quadratic functional   is the second variation of   and is denoted by,[28]

 

The second variation   is said to be strongly positive if

 

for all   and for some constant   .[29]

Using the above definitions, especially the definitions of first variation, second variation, and strongly positive, the following sufficient condition for a minimum of a functional can be stated.

Sufficient condition for a minimum:

The functional   has a minimum at   if its first variation   at   and its second variation   is strongly positive at  [30] [r][s]

See alsoEdit

NotesEdit

  1. ^ Whereas elementary calculus is about infinitesimally small changes in the values of functions without changes in the function itself, calculus of variations is about infinitesimally small changes in the function itself, which are called variations.[1]
  2. ^ See Harold J. Kushner (2004): regarding Dynamic Programming, "The calculus of variations had related ideas (e.g., the work of Caratheodory, the Hamilton-Jacobi equation). This led to conflicts with the calculus of variations community."
  3. ^ The neighborhood of   is the part of the given function space where   over the whole domain of the functions, with   a positive number that specifies the size of the neighborhood.[10]
  4. ^ Note the difference between the terms extremal and extremum. An extremal is a function that makes a functional an extremum.
  5. ^ For a sufficient condition, see section Variations and sufficient condition for a minimum.
  6. ^ The following derivation of the Euler–Lagrange equation corresponds to the derivation on pp. 184–185 of Courant & Hilbert (1953).[14]
  7. ^ Note that   and   are evaluated at the same values of   which is not valid more generally in variational calculus with non-holonomic constraints.
  8. ^ The product   is called the first variation of the functional   and is denoted by   Some references define the first variation differently by leaving out the   factor.
  9. ^ Note that assuming y is a function of x loses generality; ideally both should be a function of some other parameter. This approach is good solely for instructive purposes.
  10. ^ As a historical note, this is an axiom of Archimedes. See e.g. Kelland (1843).[15]
  11. ^ The resulting controversy over the validity of Dirichlet's principle is explained by Turnbull.[21]
  12. ^ The first variation is also called the variation, differential, or first differential.
  13. ^ The second variation is also called the second differential.
  14. ^ Note that   and the variations below, depend on both   and   The argument   has been left out to simplify the notation. For example,   could have been written  [23]
  15. ^ A functional   is said to be linear if     and     where   are functions and   is a real number.[24]
  16. ^ For a function   that is defined for   where   and   are real numbers, the norm of   is its maximum absolute value, i.e.  [25]
  17. ^ A functional is said to be quadratic if it is a bilinear functional with two argument functions that are equal. A bilinear functional is a functional that depends on two argument functions and is linear when each argument function in turn is fixed while the other argument function is variable.[27]
  18. ^ For other sufficient conditions, see in Gelfand & Fomin 2000,
    • Chapter 5: "The Second Variation. Sufficient Conditions for a Weak Extremum" – Sufficient conditions for a weak minimum are given by the theorem on p. 116.
    • Chapter 6: "Fields. Sufficient Conditions for a Strong Extremum" – Sufficient conditions for a strong minimum are given by the theorem on p. 148.
  19. ^ One may note the similarity to the sufficient condition for a minimum of a function, where the first derivative is zero and the second derivative is positive.

ReferencesEdit

  1. ^ a b Courant & Hilbert 1953, p. 184
  2. ^ Gelfand, I. M.; Fomin, S. V. (2000). Silverman, Richard A. (ed.). Calculus of variations (Unabridged repr. ed.). Mineola, New York: Dover Publications. p. 3. ISBN 978-0486414485.
  3. ^ a b Thiele, Rüdiger (2007). "Euler and the Calculus of Variations". In Bradley, Robert E.; Sandifer, C. Edward (eds.). Leonhard Euler: Life, Work and Legacy. Elsevier. p. 249. ISBN 9780080471297.
  4. ^ Goldstine, Herman H. (2012). A History of the Calculus of Variations from the 17th through the 19th Century. Springer Science & Business Media. p. 110. ISBN 9781461381068.
  5. ^ a b c van Brunt, Bruce (2004). The Calculus of Variations. Springer. ISBN 978-0-387-40247-5.
  6. ^ a b Ferguson, James (2004). "Brief Survey of the History of the Calculus of Variations and its Applications". arXiv:math/0402357.
  7. ^ Dimitri Bertsekas. Dynamic programming and optimal control. Athena Scientific, 2005.
  8. ^ Bellman, Richard E. (1954). "Dynamic Programming and a new formalism in the calculus of variations". Proc. Natl. Acad. Sci. 40 (4): 231–235. Bibcode:1954PNAS...40..231B. doi:10.1073/pnas.40.4.231. PMC 527981. PMID 16589462.
  9. ^ "Richard E. Bellman Control Heritage Award". American Automatic Control Council. 2004. Retrieved 2013-07-28.
  10. ^ Courant, R; Hilbert, D (1953). Methods of Mathematical Physics. I (First English ed.). New York: Interscience Publishers, Inc. p. 169. ISBN 978-0471504474.
  11. ^ Gelfand & Fomin 2000, pp. 12–13
  12. ^ Gelfand & Fomin 2000, p. 13
  13. ^ Gelfand & Fomin 2000, pp. 14–15
  14. ^ Courant, R.; Hilbert, D. (1953). Methods of Mathematical Physics. I (First English ed.). New York: Interscience Publishers, Inc. ISBN 978-0471504474.
  15. ^ Kelland, Philip (1843). Lectures on the principles of demonstrative mathematics. p. 58 – via Google Books.
  16. ^ Weisstein, Eric W. "Euler–Lagrange Differential Equation". mathworld.wolfram.com. Wolfram. Eq. (5).
  17. ^ Kot, Mark (2014). "Chapter 4: Basic Generalizations". A First Course in the Calculus of Variations. American Mathematical Society. ISBN 978-1-4704-1495-5.
  18. ^ Manià, Bernard (1934). "Sopra un esempio di Lavrentieff". Bollenttino dell'Unione Matematica Italiana. 13: 147–153.
  19. ^ Ball & Mizel (1985). "One-dimensional Variational problems whose Minimizers do not satisfy the Euler-Lagrange equation". Archive for Rational Mechanics and Analysis. 90 (4): 325–388. Bibcode:1985ArRMA..90..325B. doi:10.1007/BF00276295. S2CID 55005550.
  20. ^ Ferriero, Alessandro (2007). "The Weak Repulsion property". Journal de Mathématiques Pures et Appliquées. 88 (4): 378–388. doi:10.1016/j.matpur.2007.06.002.
  21. ^ Turnbull. "Riemann biography". UK: U. St. Andrew.
  22. ^ Gelfand & Fomin 2000, pp. 11–12, 99
  23. ^ Gelfand & Fomin 2000, p. 12, footnote 6
  24. ^ Gelfand & Fomin 2000, p. 8
  25. ^ Gelfand & Fomin 2000, p. 6
  26. ^ Gelfand & Fomin 2000, pp. 11–12
  27. ^ Gelfand & Fomin 2000, pp. 97–98
  28. ^ Gelfand & Fomin 2000, p. 99
  29. ^ Gelfand & Fomin 2000, p. 100
  30. ^ Gelfand & Fomin 2000, p. 100, Theorem 2

Further readingEdit

External linksEdit