Английская Википедия:Cauchy–Riemann equations

A visual depiction of a vector X in a domain being multiplied by a complex number z, then mapped by f, versus being mapped by f then being multiplied by z afterwards. If both of these result in the point ending up in the same place for all X and z, then f satisfies the Cauchy–Riemann condition.

Шаблон:Complex analysis sidebar In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which form a necessary and sufficient condition for a complex function of a complex variable to be complex differentiable.

These equations are Шаблон:NumBlk and Шаблон:NumBlk where Шаблон:Math and Шаблон:Math are real differentiable bivariate functions.

Typically, Шаблон:Math and Шаблон:Math are respectively the real and imaginary parts of a complex-valued function Шаблон:Math of a single complex variable Шаблон:Math where Шаблон:Math and Шаблон:Math are real variables; Шаблон:Math and Шаблон:Math are real differentiable functions of the real variables. Then Шаблон:Math is complex differentiable at a complex point if and only if the partial derivatives of Шаблон:Math and Шаблон:Math satisfy the Cauchy–Riemann equations at that point.

A holomorphic function is a complex function that is differentiable at every point of some open subset of the complex plane Шаблон:Math. It has been proved that holomorphic functions are analytic and analytic complex functions are complex-differentiable. In particular, holomorphic functions are infinitely complex-differentiable.

This equivalence between differentiability and analyticity is the starting point of all complex analysis.

History

The Cauchy–Riemann equations first appeared in the work of Jean le Rond d'Alembert.Шаблон:R Later, Leonhard Euler connected this system to the analytic functions.Шаблон:R CauchyШаблон:R then used these equations to construct his theory of functions. Riemann's dissertation on the theory of functions appeared in 1851.Шаблон:R

Simple example

Suppose that <math>z = x + iy</math>. The complex-valued function <math>f(z) = z^2</math> is differentiable at any point Шаблон:Mvar in the complex plane. <math display="block">f(z) = (x + iy)^2 = x^2 - y^2 + 2ixy</math> The real part <math>u(x,y)</math> and the imaginary part <math>v(x, y)</math> are <math display="block">\begin{align}

 u(x, y) &= x^2 - y^2 \\
 v(x, y) &= 2xy

\end{align}</math> and their partial derivatives are <math display="block">u_x = 2x;\quad u_y = -2y;\quad v_x = 2y;\quad v_y = 2x</math>

We see that indeed the Cauchy–Riemann equations are satisfied, <math>u_x = v_y</math> and <math>u_y = -v_x</math>.

Interpretation and reformulation

The Cauchy-Riemann equations are one way of looking at the condition for a function to be differentiable in the sense of complex analysis: in other words, they encapsulate the notion of function of a complex variable by means of conventional differential calculus. In the theory there are several other major ways of looking at this notion, and the translation of the condition into other language is often needed.

Conformal mappings

Шаблон:Further

First, the Cauchy–Riemann equations may be written in complex form Шаблон:NumBlk

In this form, the equations correspond structurally to the condition that the Jacobian matrix is of the form <math display="block">\begin{pmatrix}

 a & -b  \\
 b &  a

\end{pmatrix},</math> where <math> a = \partial u/\partial x = \partial v/\partial y</math> and <math> b = \partial v/\partial x = -\partial u/\partial y</math>. A matrix of this form is the matrix representation of a complex number. Geometrically, such a matrix is always the composition of a rotation with a scaling, and in particular preserves angles. The Jacobian of a function Шаблон:Math takes infinitesimal line segments at the intersection of two curves in z and rotates them to the corresponding segments in Шаблон:Math. Consequently, a function satisfying the Cauchy–Riemann equations, with a nonzero derivative, preserves the angle between curves in the plane. That is, the Cauchy–Riemann equations are the conditions for a function to be conformal.

Moreover, because the composition of a conformal transformation with another conformal transformation is also conformal, the composition of a solution of the Cauchy–Riemann equations with a conformal map must itself solve the Cauchy–Riemann equations. Thus the Cauchy–Riemann equations are conformally invariant.

Complex differentiability

Let <math display="block"> f(z) = u(z) + i \cdot v(z) </math> where <math display="inline">u</math> and <math>v</math> are real-valued functions, be a complex-valued function of a complex variable <math display="inline"> z = x + i y</math> where <math display="inline"> x</math> and <math display="inline"> y</math> are real variables. <math display="inline"> f(z) = f(x + iy) = f(x,y)</math> so the function can also be regarded as a function of real variables <math display="inline"> x</math> and <math display="inline"> y</math>. Then, the complex-derivative of <math display="inline"> f </math> at a point <math display="inline"> z_0=x_0+iy_0 </math> is defined by <math display="block"> f'(z_0) =\lim_{\underset{h\in\Complex}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} </math> provided this limit exists (that is, the limit exists along every path approaching <math display="inline"> z_{0} </math>, and does not depend on the chosen path).

A fundamental result of complex analysis is that <math>f</math> is complex differentiable at <math>z_0</math> (that is, it has a complex-derivative), if and only if the bivariate real functions <math>u(x+iy)</math> and <math>v(x+iy)</math> are differentiable at <math>(x_0,y_0),</math> and satisfy the Cauchy–Riemann equations at this point.Шаблон:Sfn Шаблон:Sfn^[1]

In fact, if the complex derivative exists at <math display="inline"> z_0</math>, then it may be computed by taking the limit at <math display="inline"> z_0</math> along the real axis and the imaginary axis, and the two limits must be equal. Along the real axis, the limit is <math display="block">\lim_{\underset{h\in\Reals}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} = \left. \frac{\partial f}{\partial x} \right \vert_{z_0}</math> and along the imaginary axis, the limit is <math display="block">\lim_{\underset{h\in \Reals}{h\to 0}} \frac{f(z_0+ih)-f(z_0)}{ih} = \left. \frac{1}{i}\frac{\partial f}{\partial y} \right \vert _{z_0}.</math>

So, the equality of the derivatives implies <math display="block">i \left. \frac{\partial f}{\partial x} \right \vert _{z_0} = \left. \frac{\partial f}{\partial y} \right \vert _{z_0}</math> which is the complex form of Cauchy–Riemann equations at <math display="inline"> z_0</math>.

(Note that if <math>f</math> is complex differentiable at <math>z_0</math>, it is also real differentiable and the Jacobian of <math>f</math> at <math>z_0</math> is the complex scalar <math>f'(z_0)</math>, regarded as a real-linear map of <math>\mathbb C</math>, since the limit <math>|f(z)-f(z_0)-f'(z_0)(z-z_0)|/|z-z_0|\to 0</math> as <math>z\to z_0</math>.)

Conversely, if Шаблон:Mvar is differentiable at <math display="inline"> z_{0} </math> (in the real sense) and satisfies the Cauchy-Riemann equations there, then it is complex-differentiable at this point. Assume that Шаблон:Mvar as a function of two real variables Шаблон:Mvar and Шаблон:Mvar is differentiable at Шаблон:Math (real differentiable). This is equivalent to the existence of the following linear approximation <math display="block"> \Delta f(z_0) = f(z_0 + \Delta z) - f(z_0) = f_x \,\Delta x + f_y \,\Delta y + \eta(\Delta z)</math>where <math display="inline"> f_x = \left. \frac{\partial f}{\partial x}\right \vert _{z_0} </math>, <math display="inline"> f_y = \left. \frac{\partial f}{\partial y} \right \vert _{z_0} </math>, Шаблон:Math, and <math display="inline">\eta(\Delta z) / |\Delta z| \to 0</math> as Шаблон:Math.

Since <math display="inline"> \Delta z + \Delta \bar{z}= 2 \, \Delta x </math> and <math display="inline"> \Delta z - \Delta \bar{z}=2i \, \Delta y </math>, the above can be re-written as

<math display="block"> \Delta f(z_0) = \frac{f_x - if_y}{2} \, \Delta z + \frac{f_x + if_y}{2} \, \Delta \bar{z} + \eta(\Delta z)\, </math><math display="block">\frac{\Delta f}{\Delta z} = \frac{f_x -i f_y}{2}+ \frac{f_x + i f_y}{2}\cdot \frac{\Delta\bar{z}}{\Delta z} + \frac{\eta(\Delta z)}{\Delta z}, \;\;\;\;(\Delta z \neq 0). </math>

Now, if <math display="inline">\Delta z</math> is real, <math display="inline">\Delta\bar z/\Delta z = 1</math>, while if it is imaginary, then <math display="inline">\Delta\bar z/\Delta z=-1</math>. Therefore, the second term is independent of the path of the limit <math display="inline">\Delta z\to 0</math> when (and only when) it vanishes identically: <math display="inline">f_x + i f_y=0</math>, which is precisely the Cauchy–Riemann equations in the complex form. This proof also shows that, in that case, <math display="block">\left.\frac{df}{dz}\right|_{z_0} = \lim_{\Delta z\to 0}\frac{\Delta f}{\Delta z} = \frac{f_x - i f_y}{2}.</math>

Note that the hypothesis of real differentiability at the point <math>z_0</math> is essential and cannot be dispensed with. For example,^[2] the function <math>f(x,y) = \sqrt{|xy|}</math>, regarded as a complex function with imaginary part identically zero, has both partial derivatives at <math>(x_0,y_0)=(0,0)</math>, and it moreover satisfies the Cauchy–Riemann equations at that point, but it is not differentiable in the sense of real functions (of several variables), and so the first condition, that of real differentiability, is not met. Therefore, this function is not complex differentiable.

Some sources^[3]^[4] state a sufficient condition for the complex differentiability at a point <math>z_0</math> as, in addition to the Cauchy–Riemann equations, the partial derivatives of <math>u</math> and <math>v</math> be continuous at the point because this continuity condition ensures the existence of the aforementioned linear approximation. Note that it is not a necessary condition for the complex differentiability. For example, the function <math>f(z) = z^2e^{i/|z|}</math> is complex differentiable at 0, but its real and imaginary parts have discontinuous partial derivatives there. Since complex differentiability is usually considered in an open set, where it in fact implies continuity of all partial derivatives (see below), this distinction is often elided in the literature.

Independence of the complex conjugate

The above proof suggests another interpretation of the Cauchy–Riemann equations. The complex conjugate of <math>z</math>, denoted <math display="inline">\bar{z}</math>, is defined by <math display="block">\overline{x + iy} := x - iy</math> for real variables <math>x</math> and <math>y</math>. Defining the two Wirtinger derivatives as<math display="block"> \frac{\partial}{\partial z}

 = \frac{1}{2} \left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right), \;\;\; \frac{\partial}{\partial\bar{z}}
 = \frac{1}{2} \left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right),

</math> the Cauchy–Riemann equations can then be written as a single equation <math display="block">\frac{\partial f}{\partial\bar{z}} = 0,</math> and the complex derivative of <math display="inline">f</math> in that case is <math display="inline">\frac{df}{dz}=\frac{\partial f}{\partial z}.</math> In this form, the Cauchy–Riemann equations can be interpreted as the statement that a complex function <math display="inline">f</math> of a complex variable <math display="inline">z</math> is independent of the variable <math display="inline">\bar{z}</math>. As such, we can view analytic functions as true functions of one complex variable (<math display="inline">z</math>) instead of complex functions of two real variables (<math display="inline">x</math> and <math display="inline">y</math>).

Physical interpretation

Файл:Contours of holomorphic function.png

Contour plot of a pair u and v satisfying the Cauchy–Riemann equations. Streamlines (v = const, red) are perpendicular to equipotentials (u = const, blue). The point (0,0) is a stationary point of the potential flow, with six streamlines meeting, and six equipotentials also meeting and bisecting the angles formed by the streamlines.

A standard physical interpretation of the Cauchy–Riemann equations going back to Riemann's work on function theoryШаблон:R is that u represents a velocity potential of an incompressible steady fluid flow in the plane, and v is its stream function. Suppose that the pair of (twice continuously differentiable) functions u and v satisfies the Cauchy–Riemann equations. We will take u to be a velocity potential, meaning that we imagine a flow of fluid in the plane such that the velocity vector of the fluid at each point of the plane is equal to the gradient of u, defined by <math display="block">\nabla u = \frac{\partial u}{\partial x}\mathbf i + \frac{\partial u}{\partial y}\mathbf j.</math>

By differentiating the Cauchy–Riemann equations for the functions u and v, with the symmetry of second derivatives, one shows that u solves Laplace's equation: <math display="block">\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0.</math> That is, u is a harmonic function. This means that the divergence of the gradient is zero, and so the fluid is incompressible.

The function v also satisfies the Laplace equation, by a similar analysis. Also, the Cauchy–Riemann equations imply that the dot product <math display="inline">\nabla u\cdot\nabla v = 0</math> (<math display="inline">\nabla u\cdot\nabla v = \frac{\partial u}{\partial x} \cdot \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \cdot \frac{\partial v}{\partial y} = \frac{\partial u}{\partial x} \cdot \frac{\partial v}{\partial x} - \frac{\partial u}{\partial x} \cdot \frac{\partial v}{\partial x} = 0</math>), i.e., the direction of the maximum slope of u and that of v are orthogonal to each other. This implies that the gradient of u must point along the <math display="inline">v = \text{const}</math> curves; so these are the streamlines of the flow. The <math display="inline">u = \text{const}</math> curves are the equipotential curves of the flow.

A holomorphic function can therefore be visualized by plotting the two families of level curves <math display="inline">u=\text{const}</math> and <math display="inline">v=\text{const}</math>. Near points where the gradient of u (or, equivalently, v) is not zero, these families form an orthogonal family of curves. At the points where <math display="inline">\nabla u=0</math>, the stationary points of the flow, the equipotential curves of <math display="inline">u=\text{const}</math> intersect. The streamlines also intersect at the same point, bisecting the angles formed by the equipotential curves.

Harmonic vector field

Another interpretation of the Cauchy–Riemann equations can be found in Pólya & Szegő.Шаблон:R Suppose that u and v satisfy the Cauchy–Riemann equations in an open subset of R², and consider the vector field <math display="block">\bar{f} = \begin{bmatrix} u\\ -v \end{bmatrix}</math> regarded as a (real) two-component vector. Then the second Cauchy–Riemann equation (Шаблон:EquationNote) asserts that <math>\bar{f}</math> is irrotational (its curl is 0): <math display="block">\frac{\partial (-v)}{\partial x} - \frac{\partial u}{\partial y} = 0.</math>

The first Cauchy–Riemann equation (Шаблон:EquationNote) asserts that the vector field is solenoidal (or divergence-free): <math display="block">\frac{\partial u}{\partial x} + \frac{\partial (-v)}{\partial y}=0.</math>

Owing respectively to Green's theorem and the divergence theorem, such a field is necessarily a conservative one, and it is free from sources or sinks, having net flux equal to zero through any open domain without holes. (These two observations combine as real and imaginary parts in Cauchy's integral theorem.) In fluid dynamics, such a vector field is a potential flow.Шаблон:R In magnetostatics, such vector fields model static magnetic fields on a region of the plane containing no current. In electrostatics, they model static electric fields in a region of the plane containing no electric charge.

This interpretation can equivalently be restated in the language of differential forms. The pair u and v satisfy the Cauchy–Riemann equations if and only if the one-form <math>v\,dx + u\, dy</math> is both closed and coclosed (a harmonic differential form).

Preservation of complex structure

Another formulation of the Cauchy–Riemann equations involves the complex structure in the plane, given by <math display="block">J = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}.</math> This is a complex structure in the sense that the square of J is the negative of the 2×2 identity matrix: <math>J^2 = -I</math>. As above, if u(x,y) and v(x,y) are two functions in the plane, put

<math display="block">f(x,y) = \begin{bmatrix}u(x,y)\\v(x,y)\end{bmatrix}.</math>

The Jacobian matrix of f is the matrix of partial derivatives <math display="block">Df(x,y) = \begin{bmatrix} \dfrac{\partial u}{\partial x} & \dfrac{\partial u}{\partial y} \\[5pt] \dfrac{\partial v}{\partial x} & \dfrac{\partial v}{\partial y} \end{bmatrix}</math>

Then the pair of functions u, v satisfies the Cauchy–Riemann equations if and only if the 2×2 matrix Df commutes with J.Шаблон:R

This interpretation is useful in symplectic geometry, where it is the starting point for the study of pseudoholomorphic curves.

Other representations

Other representations of the Cauchy–Riemann equations occasionally arise in other coordinate systems. If (Шаблон:EquationNote) and (Шаблон:EquationNote) hold for a differentiable pair of functions u and v, then so do <math display="block">

 \frac{\partial u}{\partial n} =  \frac{\partial v}{\partial s},\quad
 \frac{\partial v}{\partial n} = -\frac{\partial u}{\partial s}

</math>

for any coordinate system Шаблон:Math such that the pair <math display="inline">(\nabla n,\nabla s)</math> is orthonormal and positively oriented. As a consequence, in particular, in the system of coordinates given by the polar representation <math>z = r e^{i\theta}</math>, the equations then take the form <math display="block">

 {\partial u \over \partial r} =  {1 \over r}{\partial v \over \partial\theta},\quad
 {\partial v \over \partial r} = -{1 \over r}{\partial u \over \partial\theta}.

</math>

Combining these into one equation for Шаблон:Math gives <math display="block">{\partial f \over \partial r} = {1 \over ir}{\partial f \over \partial\theta}.</math>

The inhomogeneous Cauchy–Riemann equations consist of the two equations for a pair of unknown functions Шаблон:Math and Шаблон:Math of two real variables <math display="block">\begin{align}

 \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} &= \alpha(x, y) \\[4pt]
 \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} &= \beta(x, y)

\end{align}</math>

for some given functions Шаблон:Math and Шаблон:Math defined in an open subset of R². These equations are usually combined into a single equation <math display="block">\frac{\partial f}{\partial\bar{z}} = \varphi(z,\bar{z})</math> where f = u + iv and 𝜑 = (α + iβ)/2.

If 𝜑 is C^k, then the inhomogeneous equation is explicitly solvable in any bounded domain D, provided 𝜑 is continuous on the closure of D. Indeed, by the Cauchy integral formula, <math display="block">f\left(\zeta, \bar{\zeta}\right) = \frac{1}{2\pi i} \iint_D \varphi\left(z, \bar{z}\right) \, \frac{dz\wedge d\bar{z}}{z - \zeta}</math> for all ζ ∈ D.

Generalizations

Goursat's theorem and its generalizations

Шаблон:See also

Suppose that Шаблон:Math is a complex-valued function which is differentiable as a function Шаблон:Math. Then Goursat's theorem asserts that f is analytic in an open complex domain Ω if and only if it satisfies the Cauchy–Riemann equation in the domain.Шаблон:Sfn In particular, continuous differentiability of f need not be assumed.Шаблон:R

The hypotheses of Goursat's theorem can be weakened significantly. If Шаблон:Math is continuous in an open set Ω and the partial derivatives of f with respect to x and y exist in Ω, and satisfy the Cauchy–Riemann equations throughout Ω, then f is holomorphic (and thus analytic). This result is the Looman–Menchoff theorem.

The hypothesis that f obey the Cauchy–Riemann equations throughout the domain Ω is essential. It is possible to construct a continuous function satisfying the Cauchy–Riemann equations at a point, but which is not analytic at the point (e.g., Шаблон:Math. Similarly, some additional assumption is needed besides the Cauchy–Riemann equations (such as continuity), as the following example illustratesШаблон:Sfn

<math display="block">f(z) = \begin{cases}

\exp\left(-z^{-4}\right) & \text{if }z \not= 0\\
                       0 & \text{if }z = 0

\end{cases}</math>

which satisfies the Cauchy–Riemann equations everywhere, but fails to be continuous at z = 0.

Nevertheless, if a function satisfies the Cauchy–Riemann equations in an open set in a weak sense, then the function is analytic. More precisely:Шаблон:Sfn

If f(z) is locally integrable in an open domain Ω ⊂ C, and satisfies the Cauchy–Riemann equations weakly, then f agrees almost everywhere with an analytic function in Ω.

This is in fact a special case of a more general result on the regularity of solutions of hypoelliptic partial differential equations.

Several variables

There are Cauchy–Riemann equations, appropriately generalized, in the theory of several complex variables. They form a significant overdetermined system of PDEs. This is done using a straightforward generalization of the Wirtinger derivative, where the function in question is required to have the (partial) Wirtinger derivative with respect to each complex variable vanish.

Complex differential forms

As often formulated, the d-bar operator <math display="block">\bar{\partial}</math> annihilates holomorphic functions. This generalizes most directly the formulation <math display="block">{\partial f \over \partial \bar z} = 0,</math> where <math display="block">{\partial f \over \partial \bar z} = {1 \over 2}\left({\partial f \over \partial x} + i{\partial f \over \partial y}\right).</math>

Bäcklund transform

Viewed as conjugate harmonic functions, the Cauchy–Riemann equations are a simple example of a Bäcklund transform. More complicated, generally non-linear Bäcklund transforms, such as in the sine-Gordon equation, are of great interest in the theory of solitons and integrable systems.

Definition in Clifford algebra

In the Clifford algebra <math>C\ell(2)</math>, the complex number <math>z = x+iy </math> is represented as <math>z \equiv x + J y</math> where <math>J \equiv \sigma_1 \sigma_2</math>, (<math>\sigma_1^2=\sigma_2^2=1, \sigma_1 \sigma_2 + \sigma_2 \sigma_1 = 0</math>, so <math>J^2=-1</math>). The Dirac operator in this Clifford algebra is defined as <math>\nabla \equiv \sigma_1 \partial_x + \sigma_2\partial_y</math>. The function <math>f=u + J v</math> is considered analytic if and only if <math>\nabla f = 0</math>, which can be calculated in the following way:

<math display="block"> \begin{align} 0 & =\nabla f= ( \sigma_1 \partial_x + \sigma_2 \partial_y )(u + \sigma_1 \sigma_2 v) \\[4pt] & =\sigma_1 \partial_x u + \underbrace{\sigma_1 \sigma_1 \sigma_2}_{=\sigma_2} \partial_x v + \sigma_2 \partial_y u + \underbrace{\sigma_2 \sigma_1 \sigma_2}_{=-\sigma_1} \partial_y v =0 \end{align} </math>

Grouping by <math>\sigma_1</math> and <math>\sigma_2</math>:

<math display="block">\nabla f = \sigma_1 ( \partial_x u - \partial_y v) + \sigma_2 ( \partial_x v + \partial_y u) = 0 \Leftrightarrow \begin{cases} \partial_x u - \partial_y v = 0\\[4pt] \partial_x v + \partial_y u = 0 \end{cases}</math>

Hence, in traditional notation:

<math display="block">\begin{cases} \dfrac{ \partial u }{ \partial x } = \dfrac{ \partial v }{ \partial y }\\[12pt] \dfrac{ \partial u }{ \partial y } = -\dfrac{ \partial v }{ \partial x } \end{cases}</math>

Conformal mappings in higher dimensions

Let Ω be an open set in the Euclidean space Rⁿ. The equation for an orientation-preserving mapping <math>f:\Omega\to\mathbb{R}^n</math> to be a conformal mapping (that is, angle-preserving) is that <math display="block">Df^\mathsf{T} Df = (\det(Df))^{2/n}I</math>

where Df is the Jacobian matrix, with transpose <math>Df^\mathsf{T}</math>, and I denotes the identity matrix.Шаблон:R For Шаблон:Math, this system is equivalent to the standard Cauchy–Riemann equations of complex variables, and the solutions are holomorphic functions. In dimension Шаблон:Math, this is still sometimes called the Cauchy–Riemann system, and Liouville's theorem implies, under suitable smoothness assumptions, that any such mapping is a Möbius transformation.

References

Шаблон:Reflist

Sources

Шаблон:Refbegin

Шаблон:Refend

External links

Шаблон:Bernhard Riemann Шаблон:Authority control

↑ Шаблон:Cite book, p. 110-112 (Translated from Russian)
↑ Шаблон:Cite book, 2.14
↑ Шаблон:Cite book
↑ Шаблон:Cite book

[1] Шаблон:Cite book, p. 110-112 (Translated from Russian)

[2] Шаблон:Cite book, 2.14

[3] Шаблон:Cite book

[4] Шаблон:Cite book

[1]

[2]

[3]

[4]

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