Is Z 2 Continuous in the Complex Plane

Cauchy-Riemann Condition

Complex Variable Theory

George B. Arfken , ... Frank E. Harris , in Mathematical Methods for Physicists (Seventh Edition), 2013

Analytic Functions

If f (z) is differentiable and single-valued in a region of the complex plane, it is said to be an analytic function in that region. ¹ Multivalued functions can also be analytic under certain restrictions that make them single-valued in specific regions; this case, which is of great importance, is taken up in detail in Section 11.6. If f (z) is analytic everywhere in the (finite) complex plane, we call it an entire function. Our theory of complex variables here is one of analytic functions of a complex variable, which points up the crucial importance of the Cauchy-Riemann conditions. The concept of analyticity carried on in advanced theories of modern physics plays a crucial role in the dispersion theory (of elementary particles). If f′(z) does not exist at z = z ₀, then z ₀ is labeled a singular point; singular points and their implications will be discussed shortly.

To illustrate the Cauchy-Riemann conditions, consider two very simple examples.

Example 11.2.1

z ² Is Analytic

Let f (z) = z ². Multiplying out (x − iy)(x − iy) = x ² − y ² + 2ixy, we identify the real part of z ² as u(x, y) = x ² − y ² and its imaginary part as v(x, y) = 2xy. Following Eq. (11.9),

$\frac{\partial u}{\partial x}=2x=\frac{\partial v}{\partial y},\frac{\partial u}{\partial y}=-2y=-\frac{\partial v}{\partial x}.$

We see that f (z) = z ² satisfies the Cauchy-Riemann conditions throughout the complex plane. Since the partial derivatives are clearly continuous, we conclude that f (z) = z ² is analytic, and is an entire function.

Example 11.2.2

z* Is Not Analytic

Let f (z) = z*, the complex conjugate of z. Now u = x and v = −y. Applying the Cauchy-Riemann conditions, we obtain

$\frac{\partial u}{\partial x}=1\ne \frac{\partial v}{\partial y}=-1.$

The Cauchy-Riemann conditions are not satisfied for any values of x or y and f (z) = z* is nowhere an analytic function of z. It is interesting to note that f (z) = z* is continuous, thus providing an example of a function that is everywhere continuous but nowhere differentiable in the complex plane.

The derivative of a real function of a real variable is essentially a local characteristic, in that it provides information about the function only in a local neighborhood, for instance, as a truncated Taylor expansion. The existence of a derivative of a function of a complex variable has much more far-reaching implications, one of which is that the real and imaginary parts of our analytic function must separately satisfy Laplace's equation in two dimensions, namely

$\frac{{\partial }^2\psi }{\partial x^2}+\frac{{\partial }^2\psi }{\partial y^2}=0.$

To verify the above statement, we differentiate the first Cauchy-Riemann equation in Eq. (11.9) with respect to x and the second with respect to y, obtaining

$\frac{{\partial }^{2}u}{\partial x^2}=\frac{{\partial }^{2}v}{\partial x\partial y},\frac{{\partial }^{2}u}{\partial y^2}=-\frac{{\partial }^{2}v}{\partial y\partial x}.$

Combining these two equations, we easily reach

(11.13) $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0,$

confirming that u (x, y), the real part of a differentiable complex function, satisfies the Laplace equation. Either by recognizing that if f (z) is differentiable, so is −i f(z) = v(x, y) − iu(x, y), or by steps similar to those leading to Eq. (11.13), we can confirm that v (x, y) also satisfies the two-dimensional (2-D) Laplace equation. Sometimes u and v are referred to as harmonic functions (not to be confused with spherical harmonics, which we will later encounter as the angular solutions to central force problems).

The solutions u (x, y) and v(x, y) are complementary in that the curves of constant u (x, y) make orthogonal intersections with the curves of constant v(x, y). To confirm this, note that if (x ₀, y ₀) is on the curve u(x, y) = c, then x ₀ + dx, y ₀ + dy is also on that curve if

$\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy=0,$

meaning that the slope of the curve of constant u at (x ₀, y ₀) is

(11.14) ${\left(\frac{dy}{dx}\right)}_u=\frac{-\partial u/\partial x}{\partial u/\partial y},$

where the derivatives are to be evaluated at (x ₀, y ₀). Similarly, we can find that the slope of the curve of constant v at (x ₀, y ₀) is

(11.15) ${\left(\frac{dy}{dx}\right)}_v=\frac{-\partial v/\partial x}{\partial v/\partial y}=\frac{\partial u/\partial y}{\partial u/\partial x},$

where the last member of Eq. (11.15) was reached using the Cauchy-Riemann equations. Comparing Eqs. (11.14) and (11.15), we note that at the same point, the slopes they describe are orthogonal (to check, verify that dx_udx_v + dy_udy_v = 0).

The properties we have just examined are important for the solution of 2-D electrostatics problems (governed by the Laplace equation). If we have identified (by methods outside the scope of the present text) an appropriate analytic function, its lines of constant u will describe electrostatic equipotentials, while those of constant v will be the stream lines of the electric field.

Finally, the global nature of our analytic function is also illustrated by the fact that it has not only a first derivative, but in addition, derivatives of all higher orders, a property which is not shared by functions of a real variable. This property will be demonstrated in Section 11.4.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780123846549000116

OVERVIEW OF COMPUTATIONAL METHODS IN ELECTROMAGNETICS

M.V.K. Chari , S.J. Salon , in Numerical Methods in Electromagnetism, 2000

2.2 GRAPHICAL METHODS

This method was first developed by Richardson [9], and was advanced by Lehman [10], Kuhlman [11], Moore [12], Stevenson and Park [13]. The procedure is fully described by Hague [14], Bewley [15], Binns and Lawrenson [16], and Moon and Spencer [17]. A summary is presented by Wright and Deutsch [18] and the following description of the method is based on their summary.

This method is called flux plotting and is based on the principle, in the electrostatic case, that the equipotential lines and electrostatic flux lines are mutually orthogonal: they cross at right angles at all points in space. Because the potential satisfies Laplace's equation in free space, we can choose analytic functions for the solution. By analytic we mean that the chosen functions satisfy the Cauchy Riemann conditions.

Let us consider a potential function ϕ that satisfies Laplace's equation. Therefore

(2.1) $\frac{{\partial }^2\varphi }{\partial x^2}+\frac{{\partial }^2\varphi }{\partial y^2}=0$

Any function z that is analytic may be chosen in terms of complex variables that satisfies equation (2.1). Therefore, we can write w as a function of z such that

(2.2) $w=F\left(z\right)=u\left(x,y\right)+jv\left(x,y\right)$

where u and v are real functions of x and y. They also satisfy the Cauchy- Riemann conditions because the function z is analytic, so that

(2.3) $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$

(2.4) $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$

To test whether u and v satisfy Laplace's equation, we differentiate (2.3) with respect to x and (2.4) with respect to y and add the resulting equations, so that

(2.5) $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=\frac{{\partial }^{2}v}{\partial x\partial y}-\frac{{\partial }^{2}v}{\partial x\partial y}=0$

Similarly, if we differentiate (2.3) with respect to y and 2.4 with respect to x and subtract the latter from the former result, we obtain

(2.6) $\frac{{\partial }^{2}u}{\partial x\partial y}-\frac{{\partial }^{2}u}{\partial x\partial y}=\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}=0$

Therefore, u and v are conjugates of each other. Bewley [15] has shown that the families of curves u(x, y) = constant and v(x, y) = constant intersect at right angles and, therefore, can be chosen to represent equipotential lines and flux lines, respectively. To obtain a graphical solution to Laplace's equation, it is necessary to subdivide the preceding orthogonal contours into equal increments Δu and Δv, respectively. The field plot can then be obtained by setting up a system of curvilinear squares or rectangles between the boundary surfaces. To illustrate this procedure, let us consider the annulus between a circular conductor at potential V and a concentric ground plane, as shown in Figure 2.1. Owing to symmetry along the x and y axes, we can consider only a quarter of the cross section.

Figure 2.1. Geometry of the Annulus

In Figure 2.2, the potential of the conductor is V and that of the ground plane is zero. It is also evident that concentric circles in the annulus and radial lines from the center of the conductor will intersect orthogonally. In this simple case, extremely accurate solutions may be obtained because of symmetry in both directions. This may not always be the case; therefore a trial-and-error method of field plotting may be necessary. Let us now consider a small region, which may be subdivided into curvilinear rectangles as shown in Figure 2.3.

Figure 2.2. Quarter-Section of the Annulus

Figure 2.3. Laplacian Region

Assuming δV is an equal subdivision of the equipotential surface or line and δψ of the flux line, we can construct curvilinear rectangles as shown in Figure 2.3. The only requirement is that in each subregion of δV and δψ, the equipotential and flux lines at each corner of the deformed rectangle A–B–C–D intersect at right angles. We may start with this rectangle and extend the procedure to other neighboring rectangles by trial and error until orthogonality is obtained at any corner. This is no doubt a cumbersome and tedious procedure, but many useful solutions were obtained in the early days of field plotting by this method. Further, this method also provided an insight into the physical meaning of the solution to Laplace's equation.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780126157604500031

Complex Analysis

Alexander S. Poznyak , in Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques, Volume 1, 2008

Theorem 17.1. (The necessary conditions of differentiability)

Let f be defined in a neighborhood of the point $z\in ℂ$ and be differentiable at z. Then

(a)

the partial derivatives

$\begin{array}{cccc}\frac{\partial }{\partial x}u(x,y),& \frac{\partial }{\partial y}u(x,y),& \frac{\partial }{\partial x}\upsilon (x,y),& \frac{\partial }{\partial y}\upsilon (x,y)\end{array}$

exist;

(b)

and the following relations hold:

(17.4) $\frac{\partial }{\partial x}u\left(x,y\right)=\frac{\partial }{\partial y}\upsilon \left(x,y\right),\text{undefined}\frac{\partial }{\partial y}u\left(x,y\right)=-\frac{\partial }{\partial x}\upsilon \left(x,y\right)$

Proof

Since $f^{\text{'}}\left(z\right)$ exists then for any given $\epsilon >0$ there exists $\text{δ}=\text{δ}\left(\epsilon \right)>0$ such that

(17.5) $\left|\frac{f\left(z+\text{Δ}z\right)-f\left(z\right)}{\text{Δ}z}-f^{\text{'}}\left(z\right)\right|<\epsilon$

whenever $0<\left|\text{Δ}z\right|<\text{δ}$ . Representing $\text{Δ}z$ as $\text{Δ}z=t{e}^{i\alpha }$ , where $t=\left|\text{Δ}z\right|$ and $\alpha =\arg \text{undefined}z$ (see (13.40)), we can see that (17.5) is fulfilled independently of α when $0<t<\text{δ}$ . Let us take α = 0. This means that $\text{Δ}z=t=\text{Δ}x$ . This implies

(17.6) $\begin{array}{ll}{f}^{\text{'}}(z)\hfill & =\underset{\text{Δ}x\to 0}{\lim }[\frac{u(x+\text{Δ}x,y)-u(x,y)}{\text{Δ}x}+i\frac{\upsilon (x+\text{Δ}x,y)-\upsilon (x,y)}{\text{Δ}x}]\hfill \\ \hfill & =\frac{\partial }{\partial x}u(x,y)+i\frac{\partial }{\partial x}\upsilon (x,y)\hfill \end{array}$

Taking $\alpha =\pi /2$ we find that $\text{Δ}z=it=i\text{Δ}y$ and, therefore,

(17.7) $\begin{array}{ll}{f}^{\text{'}}\left(z\right)\hfill & =\underset{\text{Δ}y\to 0}{\lim }\left[\frac{u\left(x,y+\text{Δ}y\right)-u\left(x,y\right)}{i\text{Δ}y}+i\frac{\upsilon \left(x,y+\text{Δ}y\right)-\upsilon \left(x,y\right)}{i\text{Δ}y}\right]\hfill \\ \hfill & =\frac{1}{i}\frac{\partial }{\partial y}u\left(x,y\right)+\frac{\partial }{\partial x}\upsilon \left(x,y\right)=-i\frac{\partial }{\partial y}u\left(x,y\right)+\frac{\partial }{\partial y}\upsilon \left(x,y\right)\hfill \end{array}$

Comparing (17.6) with (17.7) we obtain (17.4). Theorem is proven.

The conditions (17.4) are called the Cauchy-Riemann conditions. They are also known as the d'Alembert-Euler conditions. The theorem given below shows that these conditions are also sufficient to provide the differentiability.

Theorem 17.2 (The sufficient conditions of differentiability)

The Cauchy-Riemann conditions (17.4) are also sufficient for the differentiability of f (z) provided the functions u (x, y) and $\upsilon \left(x,y\right)$ are totally differentiable (all partial derivatives exist) at the considered point. The derivative $f^{\text{'}}\left(z\right)$ can be calculated as

(17.8) $f^{\text{'}}\left(z\right)=\frac{\partial }{\partial x}u\left(x,y\right)+i\frac{\partial }{\partial x}\upsilon \left(x,y\right)=\frac{\partial }{\partial y}\upsilon \left(x,y\right)-i\frac{\partial }{\partial y}u\left(x,y\right)$

Proof

By the total differentiability it follows that

$\begin{array}{ll}\text{Δ}u\hfill & :=u\left(x+\text{Δ}x,y+\text{Δ}y\right)-u\left(x,y\right)=\frac{\partial u}{\partial x}\text{Δ}x+\frac{\partial u}{\partial y}\text{Δ}y+o\left(\left|\text{Δ}z\right|\right)\hfill \\ \text{Δ}\upsilon \hfill & :=\upsilon \left(x+\text{Δ}x,y+\text{Δ}y\right)-\upsilon \left(x,y\right)=\frac{\partial \upsilon }{\partial x}\text{Δ}x+\frac{\partial \upsilon }{\partial y}\text{Δ}y+o\left(\left|\text{Δ}z\right|\right)\hfill \end{array}$

Therefore

$\begin{array}{ll}\frac{f\left(z+\text{Δ}z\right)-f\left(z\right)}{\text{Δ}z}\hfill & =\frac{\text{Δ}u+i\text{Δ}\upsilon }{\text{Δ}x+i\text{Δ}y}\hfill \\ \hfill & =\frac{\left(\frac{\partial u}{\partial x}\text{Δ}x+\frac{\partial u}{\partial y}\text{Δ}y\right)+i(\frac{\partial \upsilon }{\partial x}\text{Δ}x+\frac{\partial \upsilon }{\partial y}\text{Δ}y)+o\left(\left|\text{Δ}z\right|\right)}{\text{Δ}x+i\text{Δ}y}\hfill \end{array}$

Using now the Cauchy–Riemann conditions (17.4), the simple rearrangement gives

$\begin{array}{ll}\frac{f\left(z+\text{Δ}z\right)-f\left(z\right)}{\text{Δ}z}\hfill & =\frac{\left(\left[\frac{\partial u}{\partial x}+i\frac{\partial \upsilon }{\partial x}\right]\text{Δ}x+\left[\frac{\partial u}{\partial y}+i\frac{\partial \upsilon }{\partial y}\right]\text{Δ}y\right)+o\left(\left|\text{Δ}z\right|\right)}{\text{Δ}x+i\text{Δ}y}\hfill \\ \hfill & =\frac{\left(\left[\frac{\partial u}{\partial x}+i\frac{\partial \upsilon }{\partial x}\right]\text{Δ}x+\left[-i\frac{\partial u}{\partial y}+\frac{\partial \upsilon }{\partial y}\right]i\text{Δ}y\right)+o\left(\left|\text{Δ}z\right|\right)}{\text{Δ}x+i\text{Δ}y}\hfill \\ \hfill & =\frac{\left(\left[\frac{\partial u}{\partial x}+i\frac{\partial \upsilon }{\partial x}\right]\text{Δ}x+\left[i\frac{\partial \upsilon }{\partial x}+\frac{\partial u}{\partial x}\right]i\text{Δ}y\right)+o\left(\left|\text{Δ}z\right|\right)}{\text{Δ}x+i\text{Δ}y}\hfill \\ \hfill & =\left[\frac{\partial u}{\partial x}+i\frac{\partial \upsilon }{\partial x}\right]+o\left(1\right)\hfill \end{array}$

where $o\left(1\right)\to 0$ whenever $\left|\text{Δ}z\right|\to 0$ . So that $f^{\text{'}}\left(z\right)$ exists and is given by

which completes the proof.

Example 17.2

For the same function $f\left(z\right)=z^2$ as in Example 17.1 we have

$\begin{array}{l}\frac{\partial u}{\partial x}=\frac{\partial \upsilon }{\partial x}=2x,\text{undefined}\frac{\partial u}{\partial y}=\frac{\partial \upsilon }{\partial y}-2y\\ f^{\text{'}}\left(z\right)=2x-i2y\end{array}$

Definition 17.2

A function f(z), differentiable at each point of an open set $\mathcal{D}\subset ℂ$ , is called regular (or holomorphic ) on $\mathcal{D}$ .Sure, here we assume that we deal with a single-valued (or uniform) function since the notion of differentiability (17.3) has been introduced only for single-valued functions. If a regular function f(z) possesses a continuous derivative on $\mathcal{D}$ then it is called an analytic function . ¹

Below these definitions will be extensively used.

Example 17.3

It is easy to check that, as in real analysis,

(17.9) $\begin{array}{c}\frac{d}{dz}{e}^z=e^z,\text{undefined}\frac{d}{dz}\text{ln}z=\frac{1}{z}\\ \frac{d}{dz}\sin z=\cos z,\text{undefined}\frac{d}{dz}\cos z=-\sin z\\ \frac{d}{dz}\tan z=\frac{1}{{\cos }^{2}z},\text{undefined}\frac{d}{dz}\cot z=-\frac{1}{{\sin }^{2}z}\end{array}$

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780080446745500201

Solid Mechanics

Josef Kuneš , in Dimensionless Physical Quantities in Science and Engineering, 2012

4.3 Aeroelasticity

In aeroelasticity, the dimensionless quantities express the influence of the interaction between inertia, elasticity and aerodynamic forces. In static aeroelasticity, the dimensionless quantity relates to the problems of divergence and regressive control of a bypassed surface. In dynamic aeroelasticity, the dimensionless quantity relates to the problems of fluttering and the dynamic response of mechanical parts which are bypassed. The aeroelasticity parameter, Cauchy's, Connor's, Frueh's, Regier's and other numbers are among the well-known dimensionless quantities in aeroelasticity.

4.3.1 Aeroelasticity Parameter Ae

See the Cauchy number (aeroelasticity parameter) Cau (p. 155).

Info: [A29].

4.3.2 Aeroelasticity Stiffness N _AE

$N_{\mathrm{AE}}=\frac{2E}{\varrho w^2}$

E (Pa) – modulus of elasticity; ϱ (kg   m⁻³) – density; w (m   s⁻¹) – flow velocity.

It expresses the ratio of the structure stiffness to the aerodynamic force. It is the inverse value of the aeroelasticity parameter AE (p. 155). Aeroelasticity.

4.3.3 Cauchy Number (Aeroelasticity Parameter) Cau

$Cau\equiv M^2\equiv Ho=\frac{\varrho w^2}{K}(1);$

$Cau=\frac{\varrho w^2}{E}(2)$

ϱ (kg   m⁻³) – density; w (m   s⁻¹) – velocity; K (Pa) – volume modulus of elasticity; E (Pa) – modulus of elasticity; M (–) – Mach number (p. 73); Ho (–) – Hooke number (p. 138).

This parameter expresses the ratio of the inertia force to the material compressibility force. It characterizes the compressible fluid flow (1) and the dynamic material strain (2) by inertia forces. It is often called the aeroelasticity parameter as well. Aerodynamics, aeroelasticity, dynamics.

Info: [A23],[A43],[B127],[C37].

Augustin Louis Cauchy (21.8.1789–23.5.1857), French mathematician.

He was known for his precision and consistency in mathematics. He introduced many concepts such as the determinant, limit, continuity and convergence. He founded complex analysis and deduced the Cauchy–Riemann conditions with Riemann. He was very prolific, publishing nearly 800 works.

4.3.4 Connors Number Con

$Con=\frac{w}{fL}=K\sqrt{\frac{m\delta }{\varrho L^2}}$

w (m   s⁻¹) – mean velocity of two-phase flow; f (s⁻¹) – oscillation frequency; L (m) – characteristic length, pipe diameter; K (–) – instability factor; m (kg   m⁻¹) – length density of pipe material; δ (–) – damping decrement; ϱ (kg   m⁻³) – fluid density.

It relates to the fluid elastic vibration characteristics in the two-phase flow of fluids. For example, it relates to tubes exposed to a dynamic two-phase flow of air and water in heat exchangers. The number Con serves to determine the critical flow rate and is based on the mean flow rate, mean fluid density and damping in a two-phase flow. Aeroelasticity, two-phase flow, heat exchangers.

Info: [G28].

H. J. Connors, American scientific researcher.

Together with P.M. Moretti, Connors is among the significant personalities in the aeroelasticity field, in dynamic induced oscillation and fluttering.

Peter M. Moretti, American scientist.

His scientific work is very wide-ranging and involves the dynamics of mechanical systems, especially those of induced vibrations excited by fluid streaming. Primarily, this relates to the fluid stream dynamic acting on the nest of tubes in heat exchangers, further about the instability solution in transversal streaming through these nests and the eigenfrequency determination and damping of nests. He was also engaged in research related to thermal stratification in lakes, in supersonic streaming and heat transfer problems.

4.3.5 Flutter Number F

$F=\frac{w_{\mathrm{eq}}}{w_{\mathrm{R}}}=\frac{M}{Rg}$

w _eq (m   s⁻¹) – equivalent air velocity near the sea level; w _R (m   s⁻¹) – Regier surface velocity index (flutter parameter); M (–) – Mach number (p. 73); Rg (–) – Regier number (p. 157).

High velocities aerodynamics. Aeroelasticity. Flutter.

Info: [B94].

4.3.6 Frequency Parameter P _f

$P_f=\frac{\omega L}{w}=2\pi Sh$

ω (s⁻¹) – angular velocity; L (m) – characteristic length (e.g. channel width or distance from a wall); w (m   s⁻¹) – flow velocity; Sh (–) – Strouhal number (p. 87).

It characterizes the unsteady flow in bundles, fluidization and other dynamic processes acting on mechanical systems, as an example.

Info: [A29],[A35].

4.3.7 Frueh Number Fh

$Fh=\frac{L{\omega }_1}{a}\sqrt{\frac{N_m}{c_L}}$

L (m) – characteristic length, (e.g. half-length of wing chord); ω ₁ (s⁻¹) – first harmonic frequency of torsional vibration; a (m   s⁻¹) – sound velocity; c _L (–) – uplift curve inclination; m (kg   m⁻¹) – mass to the wing length; ϱ (kg   m⁻³) – air density; N _m (–) – mass ratio (p. 75).

Aeroelasticity, transonic flutter of wings.

Info: [A24].

Frank J. Frueh, American aerodynamic engineer.

4.3.8 Mass Ratio N _m

$N_m=\frac{m}{\pi \varrho L^3}$

m (kg) – mass of body; ϱ (kg   m⁻³) – fluid density; L (m) – characteristic length of a body.

Mechanical systems dynamics. Aircraft fluttering and stability. Aeroelasticity.

Info: [A24].

4.3.9 Regier Number Rg

$Rg=\frac{L\omega }{a}\sqrt{\frac{\varrho l}{\pi \varrho L^2}}$

L (m) – characteristic length; ω (s⁻¹) – angular velocity; a (m   s⁻¹) – sound velocity; ϱl (kg   m⁻³) – fluid density; ϱ (kg   m⁻³) – material density.

It expresses aeroelastic relations, transonic flutter of wings and blades of flow machines. It is a modification of the Strouhal number Sh (p. 87). Aeroelasticity.

Info: [A24].

4.3.10 Regier Surface Number Rg _s

$R{g}_{\mathrm{s}}=\frac{w_{\mathrm{R}}}{a}$

w _R (m   s⁻¹) – Regier surface velocity index (flutter parameter); a (m   s⁻¹) – sound velocity.

It is a simpler expression of the Regier number Rg (p. 157) and represents the elastic-to-aerodynamic forces ratio at sea level. High velocity aerodynamics. Aeroelasticity. Flutter.

Info: [A24],[B94].

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B978012416013200004X

Measure and Integration, Function Spaces

Henri Bourlès , in Fundamentals of Advanced Mathematics 2, 2018

4.2.1 Holomorphic functions in a single complex variable

Theorem-Definition 4.45

1)

Let $\Omega \subset \mathrm{ℂ}$ be an open set in the complex plane, suppose that E is a real Banach space, and write $E_{\left(\mathrm{ℂ}\right)}$ for the complexification of E ( section 3.4.1 ).

A function $\mathbf{f}:\Omega \to E_{\left(\mathrm{ℂ}\right)}:\mathbf{f}\mapsto \mathbf{f}\left(z\right)$ is said to be holomorphic at the point z ₀ ∈ Ω if the "complex derivative"

$\frac{d\mathbf{f}}{\mathit{dz}}\left(z_0\right)≔\underset{h\to 0,h\ne 0}{lim}\frac{\mathbf{f}\left(z_0+h\right)-\mathbf{f}\left(z_0\right)}{h}$

(where h is a small increment in $\mathrm{ℂ}$ ) exists in $E_{\left(\mathrm{ℂ}\right)}$ .

2)

By viewing Ω as an open subset of $\mathrm{ℝ}\times \mathrm{ℝ}$ , the function f  = U  + iV ( $U=\mathrm{\Re }\left(\mathbf{f}\right),V=\mathrm{\Im }\left(\mathbf{f}\right)$ ) defined above can be viewed as a function of the two real variables x and y, with $x=\mathrm{\Re }\left(z\right)$ and $y=\mathfrak{J}\left(z\right)$ . The following conditions are equivalent:

i)

f is holomorphic.

ii)

f : (x, y) ↦ f (x, y) is differentiable and satisfies the Cauchy-Riemann condition

[4.9] $\frac{\partial \mathbf{f}}{\partial x}+i\frac{\partial \mathbf{f}}{\partial y}=0,\mathit{or\hspace{0.17em}equivalently}\frac{\partial U}{\partial x}=\frac{\partial V}{\partial y},\frac{\partial U}{\partial y}=-\frac{\partial V}{\partial x}.$

Proof

Let z ₀ = x ₀ + iy ₀ ∈ Ω. By setting $c=\frac{d\mathbf{f}}{\mathit{dz}}\left(z_0\right)$ , condition (i) means that f (z ₀ + h) = f (z ₀) + h.c + o (h), where h is a small complex increment. The function f : (x, y) ↦ f (x, y) is differentiable at the point (x ₀, y ₀) with partial derivatives $a=\frac{\partial \mathbf{f}}{\partial x}\left(x_0,y_0\right)$ , $b=\frac{\partial \mathbf{f}}{\partial y}\left(x_0,y_0\right)$ if and only if

$\mathbf{f}\left(x_0+k,y_0+l\right)=\mathbf{f}\left(x_0,y_0\right)+k.a+l.b+o\left(\left|\left(k,l\right)\right|\right),$

where k, l are small real increments. Write h = k + il, so that h.c = (k + il) c.

Then

$h.c=k.a+l.b\iff \left(a=c,b=\mathit{ic}\right)\Rightarrow a+\mathit{ib}=0,$

so (i)⇔(ii).

In Theorem-Definition 4.45(2), z = x + iy and $\overline{z}=x-\mathit{iy}$ are viewed as independent variables. Since $x=\frac{z+\overline{z}}{2}$ , $y=\frac{z-\overline{z}}{2i}$ , we have that $\frac{\partial }{\partial \overline{z}}=\frac{\partial }{\partial x}.\frac{\partial x}{\partial \overline{z}}+\frac{\partial }{\partial y}.\frac{\partial y}{\partial \overline{z}}=\frac{1}{2}\left(\frac{\partial }{\partial x}+i\frac{\partial }{\partial y}\right)$ .

Corollary 4.46

The Cauchy-Riemann equation (4.9) is equivalent to $\frac{\partial \mathbf{f}}{\partial \overline{z}}=0$ . If f is continuous on Ω and differentiable on Ω − D, where D is finite, then this condition is satisfied on Ω − D if and only if the differential form ω = f.dz is closed, i.e. dω = 0 ⁶ .

Proof

We have that $\frac{\partial \mathbf{f}}{\partial x}+i\frac{\partial \mathbf{f}}{\partial y}=\frac{\partial \mathbf{f}}{\partial \overline{z}}$ and $\mathit{d\omega }=\frac{\partial \mathbf{f}}{\partial \overline{z}}d\overline{z}\wedge \mathit{dz}=2i\frac{\partial \mathbf{f}}{\partial \overline{z}}\mathit{dx}\wedge \mathit{dy}$ , so dω = 0 if and only if $\frac{\partial \mathbf{f}}{\partial \overline{z}}=0$ .

Clearly, the set of holomorphic functions on an open subset Ω of $\mathrm{ℂ}$ with values in $E_{\left(\mathrm{ℂ}\right)}$ is a $\mathrm{ℂ}$ -vector space. This vector space is written $\mathrm{\mathscr{H}}\left(\Omega ;E_{\left(\mathrm{ℂ}\right)}\right)$ , or simply $\mathrm{\mathscr{H}}\left(\Omega \right)$ if $E_{\left(\mathrm{ℂ}\right)}=\mathrm{ℂ}$ .

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9781785482496500043

Introduction to complex analysis

Brent J. Lewis , ... Andrew A. Prudil , in Advanced Mathematics for Engineering Students, 2022

10.2 Complex integration

If $f(x)$ is defined, single-valued, and continuous in ℜ, the integral of $f(z)$ can be defined along some path C in ℜ from point $z_1(=x_1+i{y}_1)$ to point $z_2(=x_2+i{y}_2)$ as

${\int }_{C}f(z)dz={\int }_{(x_1,y_1)}^{(x_2,y_2)}(u+iv)(dx+idy)={\int }_{(x_1,y_1)}^{(x_2,y_2)}(udx-vdy)+i{\int }_{(x_1,y_1)}^{(x_2,y_2)}(vdx+udy).$

With this definition, the integral of a function of a complex variable can be made to depend on line integrals of real functions. The rules for complex integration are similar to those for real integrals. For instance,

(10.4)

where M is an upper bound of $|f(z)|$ on C, that is, $|f(z)|⩽M$ , and L is the length of the path C.

Cauchy's theorem

Let C be a simple closed curve. If $f(z)$ is analytic within the region bounded by C as well as on C, then Cauchy's theorem states

(10.5)

where the second integral emphasizes the fact that C is a simple closed curve. Eq. (10.5) is equivalent to the statement that ${\int }_{z_1}^{z_2}f(z)dz$ has a value independent of the path joining $z_1$ and $z_2$ . Such integrals can be evaluated as $F(z_2)-F(z_1)$ , where $F^{\prime }(z)=f(z)$ . For example:

(i)

Since $f(z)=2z$ is analytic everywhere, for any simple closed curve $C:{\oint }_{c}2zdz=0$ .

(ii)

We have ${\int }_{2i}^{1+i}2zdz=z^2{|}_{2i}^{1+i}={(1+i)}^2-{(2i)}^2=2i+4$ .

Example 10.2.1

Derive Cauchy's theorem from first principles.

Solution.

Eq. (10.5) is a line integral, where $f(z)=u+iv$ is analytic over a region bounded by the closed C and dz is an infinitesimal part of the path around C such that $dz=dx+idy$ . Hence, Eq. (10.5) can be written as

(10.6) ${\oint }_{C}fz)dz=\oint (u+iv)(dx+idy)={\oint }_C(udx-vdy)+i{\oint }_C(vdx+udy).$

Consider the Green's theorem from Eq. (4.13) in Section 4.4.2:

${\oint }_C(Pdx+Qdy)={\iint }_A(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dxdy,$

where $P(x)$ and $Q(x)$ are well-behaved functions and A is the area bounded by the contour C. One can apply this theorem to each of the terms in the last expression of Eq. (10.6). Therefore, for the first term, letting $P=u$ and $Q=-v$ yields

${\oint }_C(udx-vdy)={\iint }_A(-\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y})dxdy=0.$

This expression equals zero because of the second Cauchy–Riemann condition in Eq. (10.2). Similarly, letting $P=v$ and $Q=u$ in the second term yields

${\oint }_C(vdx+udy)={\iint }_A(\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y})dxdy=0.$

This expression also equals zero because of the first Cauchy–Riemann condition in Eq. (10.2). Thus, Eq. (10.5) is shown to equal zero.   [answer]

Cauchy's integral formulas

If $f(z)$ is analytic within and on a simple closed curve C and a is any point interior to C, then

(10.7)

where C is traversed in the positive counterclockwise sense. Also the nth derivative of $f(z)$ at $z=a$ is given by

(10.8)

Eqs. (10.7) and (10.8) are called Cauchy's integral formulas. These are quite remarkable because they show that if the function $f(z)$ is known on the closed curve C, then it is also known within C, and the various derivatives and points within C can be calculated. Thus, if a function of a complex variable has a first derivative, it has all higher derivatives as well. This result of course is not necessarily true for functions of real variables.

Example 10.2.2

Evaluate ${\oint }_C\frac{\cos z}{z-\pi }dz$ , where C is the circle $|z-1|=3$ .

Solution.

Since $z=\pi$ lies within C, $\frac{1}{2\pi i}{\oint }_C\frac{\cos z}{z-\pi }dz=\cos \pi =-1$ by Eq. (10.7) with $f(z)=\cos z$ and $z=\pi$ . Therefore,

${\oint }_C\frac{\cos z}{z-\pi }dz=-2\pi i.\text{[answer]}$

Example 10.2.3

Derive the Cauchy integral formula in Eq. (10.7) from first principles.

Solution.

Consider the function $\varphi (z)=\frac{f(z)}{(z-a)}$ , which is analytic inside of C except at the point a, which lies within C, where there is a singularity. Furthermore, let $C^{\prime }$ be a small circle of radius ρ that is centered at a and make a cut between C and $C^{\prime }$ along AB, as shown in Fig. 10.1. Furthermore, consider the path in Fig. 10.1 from A around C to $A^{\prime }$ , along $A^{\prime }{B}^{\prime }$ , around $C^{\prime }$ from $B^{\prime }$ to B, and back along BA.

Figure 10.1. Schematic of a closed path for evaluation of Cauchy's integral formula.

From Cauchy's theorem in Eq. (10.5),

${\oint }_{A{A}^{\prime }{B}^{\prime }BA}\varphi (z)dz=0$

for the closed path. Since $\varphi (z)$ is analytic in the region between C and $C^{\prime }$ , we have

${\int }_{A{A}^{\prime }}\varphi (z)dz+{\int }_{A^{\prime }{B}^{\prime }}\varphi (z)dz+{\int }_{B^{\prime }B}\varphi (z)dz+{\int }_{BA}\varphi (z)dz=0.$

If now both A and $A^{\prime }$ , and B and $B^{\prime }$ each come together and coincide, then the integral along the straight line $A^{\prime }{B}^{\prime }$ is equal but opposite to the integral along BA so that these two integrals cancel, leaving the following two integrals:

${\int }_{A{A}^{\prime }}\varphi (z)dz+{\int }_{B^{\prime }B}\varphi (z)dz=0.$

Recognizing the direction of the path integrals,

$\underset{counterclockwise}{\underbrace{{\oint }_C\varphi (z)dz}}+\underset{clockwise}{\underbrace{{\oint }_{C^{\prime }}\varphi (z)dz}}=0$

or

(10.9) ${\oint }_C\varphi (z)dz-{\oint }_{C^{\prime }}\varphi (z)dz=0,$

where the integrals in Eq. (10.9) are understood to be counterclockwise. For the circle $C^{\prime }$ of radius ρ centered at a in Fig. 10.1, one can use polar coordinates so that $z=a+\rho e^{i\theta }$ and $dz=i\rho e^{i\theta }d\theta$ , and therefore

${\int }_{C^{\prime }}\varphi (z)dz={\oint }_{C^{\prime }}\frac{f(z)}{(z-a)}dz={\int }_0^{2\pi }\frac{f(z)}{\rho e^{i\theta }}i\rho e^{i\theta }d\theta =i{\int }_0^{2\pi }f(z)d\theta .$

Since this result is valid for any ρ, one can let ρ approach zero for which z approaches a. As $f(z)$ is both continuous and analytic at $z=a$ , in the limit $\underset{z\to a}{\lim }f(z)=f(a)$ . Thus, the integral around $C^{\prime }$ is evaluated as

${\oint }_{C^{\prime }}\varphi (z)dz=if(a){\int }_0^{2\pi }d\theta =2\pi if(a).$

Hence, substituting this result into Eq. (10.9) yields the final result in Eq. (10.7):

$f(a)=\frac{1}{2\pi i}{\oint }_C\frac{f(z)}{(z-a)}dz.\text{[answer]}$

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128236819000186

ELECTROSTATICS OF CONDUCTORS

L.D. LANDAU , E.M. LIFSHITZ , in Electrodynamics of Continuous Media (Second Edition), 1984

(3) The method of conformai mapping

A field which depends on only two Cartesian coordinates (x and y, say) is said to be two-dimensional. The theory of functions of a complex variable is a powerful means of solving two-dimensional problems of electrostatics. The theoretical basis of the method is as follows.

An electrostatic field in a vacuum satisfies two equations: curl E = 0, div E = 0. The first of these makes it possible to introduce the field potential, defined by E = − grad ϕ. The second equation shows that we can also define a vector potential A of the field, such that E = curl A. In the two-dimensional case, the vector E lies in the xy-plane, and depends only on x and y. Accordingly, the vector A can be chosen so that it is perpendicular to the xy-plane. Then the field components are given in terms of the derivatives of ϕ and A by

(3.14) $\begin{array}{cc}{E}_x=-\partial \varphi \text{/}\partial x=\partial A\text{/}\partial y,& E_y=-\partial \varphi \text{/}\partial y=\partial A\text{/}\partial x.\end{array}$

These relations between the derivatives of ϕ and A are, mathematically, just the well-known Cauchy–Riemann conditions, which express the fact that the complex quantity

(3.15) $w=\varphi -iA$

is an analytic function of the complex argument z = x + iy. This means that the function w(z) has a definite derivative at every point, independent of the direction in which the derivative is taken. For example, differentiating along the x-axis, we find dw/dz = ∂ϕ/∂x − i∂A/∂x, or

(3.16) $\text{d}w\text{/d}z=-E_x+i{E}_y.$

The function w is called the complex potential.

The lines of force are defined by the equation dx/E_x = dy/E_y. Expressing E_x and E_y as derivatives of A, we can write this as (∂A/∂x)dx + (∂A/∂y)dy = dA = 0, whence A(x, y) = constant. Thus the lines on which the imaginary part of the function w(z) is constant are the lines of force. The lines on which its real part is constant are the equipotential lines. The orthogonality of these families of lines is ensured by the relations (3.14), according to which

$\frac{\partial \varphi }{\partial x}\frac{\partial A}{\partial x}+\frac{\partial \varphi }{\partial y}\frac{\partial A}{\partial y}=0.$

Both the real and the imaginary part of an analytic function w(z) satisfy Laplace's equation. We could therefore equally well take im w as the field potential. The lines of force would then be given by re w = constant. Instead of (3.15) we should have w = A + iϕ.

The flux of the electric field through any section of an equipotential line is given by the integral ∮E_n dl = − ∮(∂ϕ/∂n)dl, where dl is an element of length of the equipotential line and n the direction of the normal to it. According to (3.14) we have ∂n/∂Π = − ∂A/∂l, the choice of sign denoting that l is measured to the left when one looks along n. Thus ∮ E_n dl = ∮(∂A/∂l)dl = A ₂ − A ₁, where A ₂ and A ₁ are the values of A at the ends of the section. In particular, since the flux of the electric field through a closed contour is 4πe, where e is the total charge enclosed by the contour (per unit length of conductors perpendicular to the plane), it follows that

(3.17) $e=\left(1\text{/}4\pi \right)\Delta A,$

where ΔA is the change in A on passing counterclockwise round the closed equipotential line.

The simplest example of the complex potential is that of the field of a charged straight wire passing through the origin and perpendicular to the plane. The field is given by E_r = 2e/r, E_θ = 0, where r, θ are polar coordinates in the xy-plane, and e is the charge per unit length of the wire. The corresponding complex potential is

(3.18) $w=-2e\log z=-2e\log r-2ie\theta .$

If the charged wire passes through the point (x ₀, y ₀) instead of the origin, the complex potential is

(3.19) $w=-2e\log \left(z-z_0\right),$

where z ₀ = x ₀ + iy ₀.

Mathematically, the functional relation w = w(z) constitutes a conformai mapping of the plane of the complex variable z on the plane of the complex variable w. Let C be the cross-sectional contour of a conductor in the xy-plane, and ϕ₀ its potential. It is clear from the above discussion that the problem of determining the field due to this conductor amounts to finding a function w(z) which maps the contour C in the z-plane on the line w = ϕ₀, parallel to the axis of ordinates, in the w-plane. Then re w gives the potential of the field. (If the function w(z) maps the contour C on a line parallel to the axis of abscissae, then the potential is im w.)

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780080302751500072

seaveywhisingerrim1994.blogspot.com

Source: https://www.sciencedirect.com/topics/mathematics/cauchy-riemann-condition