Chapter 12 Applications of Analytic Functions

12.1 Properties of Analytic Functions Complex valued functions of a single complex variable are useful in various disciplines such as physics and numerical approximation theory. The current chapter summarizes a number of attractive properties of analytic functions and presents some applications in which MATLAB is helpful. Excellent textbooks presenting the theory of analytic functions [18, 75, 119] are available which fully develop various theoretical concepts employed in this chapter. Therefore, only the properties which may be helpful in subsequent discussions are included.

12.2 DeÞnition of Analyticity We consider a complex valued function F (z) = u(x, y) + iv(x, y)

,

z = x + iy

which depends on the complex variable z. The function F (z) is analytic at point z if it is differentiable in the neighborhood of z. Differentiability requires that the limit

F (z + ∆z) − F (z) lim = F  (z) ∆z |∆z|→0 exists independent of how |∆z| approaches zero. Necessary and sufÞcient conditions for analyticity are continuity of the Þrst partial derivatives of u and v and satisfaction of the Cauchy-Riemann conditions (CRC) ∂u ∂v = ∂x ∂y

,

∂u ∂v =− ∂y ∂x

These conditions can be put in more general form as follows. Let n denote an arbitrary direction in the z-plane and let s be the direction obtained by a 90 ◦ counterclockwise rotation from the direction of n. The generalized CRC are: ∂v ∂u = ∂n ∂s

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,

∂u ∂v =− ∂s ∂n

Satisfaction of the CRC implies that both u and v are solutions of Laplace’s equation ∇2 u =

∂2u ∂2u + 2 =0 ∂x2 ∂y

and

∂2v ∂2v + 2 =0 2 ∂x ∂y These functions are called harmonic. Functions related by the CRC are also said to be harmonic conjugates. When one function u is known, its harmonic conjugate v can be found within an additive constant by using       ∂v ∂u ∂u ∂v dx + dy = − dx + dy + constant v = dv = ∂x ∂y ∂y ∂x Harmonic conjugates also have the properties that curves u = constant and v = ∂u constant intersect orthogonally. This follows because u = constant implies ∂n is ∂u ∂v zero in a direction tangent to the curve. However ∂n = ∂s so v = constant along a curve intersecting u = constant orthogonally. Sometimes it is helpful to regard a function of x and y as a function of z = x + iy and z¯ = x − iy. The inverse is x = (z + z¯)/2 and y = (z − z¯)/(2i). Chain rule differentiation applied to a general function φ yields ∂φ ∂φ ∂φ = + ∂x ∂z ∂ z¯ so that



∂ ∂ −i ∂x ∂y

 φ=2

∂φ ∂z

∂φ ∂φ ∂φ =i −i ∂y ∂z ∂ z¯

,

 ,

∂ ∂ +i ∂x ∂y

 φ=2

∂φ ∂ z¯

So Laplace’s equation becomes ∂2φ ∂2φ ∂2φ =0 + 2 =4 2 ∂x ∂y ∂z∂ z¯ It is straightforward to show the condition that a function F be an analytic function of z is expressible as ∂F =0 ∂ z¯ It is important to note that most of the functions routinely employed with real arguments are analytic in some part of the z-plane. These include: √ z n , z, log(z), ez , sin(z), cos(z), arctan(z), to mention a few. The real and imaginary parts of these functions are harmonic and they arise in various physical applications. The integral powers of z are especially signiÞcant. We can write

y z = reıθ , r = x2 + y 2 , θ = tan−1 x

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and get z n = u + iv

u = rn cos(nθ) ,

,

v = rn sin(nθ)

The reader can verify by direct differentiation that both u and v are harmonic. Points where F (z) is nondifferentiable are called singular points and these are categorized as isolated or nonisolated. Isolated singularities are termed either poles or essential singularities. Branch points are the most common type of nonisolated singularity. Singular points and their signiÞcance are discussed further below.

12.3 Series Expansions If F (z) is analytic inside and on the boundary of an annulus deÞned by a ≤ |z − z0 | ≤ b then F (z) is representable in a Laurent series of the form F (z) =

∞ 

an (z − z0 )n

a ≤ |z − z0 | ≤ b

,

n=−∞

where

 F (t) dt 1 F (z) = 2πı L (t − z0 )n+1 and L represents any closed curve encircling z 0 and lying between the inner circle |z − z0 | = a and the outer circle |z − z 0 | = b. The direction of integration along the curve is counterclockwise. If F (z) is also analytic for |z − z 0 | < a, the negative powers in the Laurent series drop out to give Taylor’s series F (z) =

∞ 

an (z − z0 )n

,

|z − z0 | ≤ b

n=0

Special cases of the Laurent series lead to classiÞcation of isolated singularities as poles or essential singularities. Suppose the inner radius can be made arbitrarily small but nonzero. If the coefÞcients below some order, say −m, vanish but a −m = 0, we classify z0 as a pole of order m. Otherwise, we say z 0 is an essential singularity. Another term of importance in connection with Laurent series is a −1 , the coefÞcient of (z − z0 )−1 . This coefÞcient, called the residue at z 0 , is sometimes useful for evaluating integrals.

12.4 Integral Properties Analytic functions have many useful integral properties. One of these properties that concerns integrals around closed curves is:

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Cauchy-Goursat Theorem: If F (z) is analytic at all points in a simply connected region R, then  F (z) dz = 0 L

for every closed curve L in the region. An immediate consequence of this theorem is that the integral of F (z) along any path between two end points z 1 and z2 is independent of the path (this only applies for simply connected regions).

12.4.1 Cauchy Integral Formula If F (z) is analytic inside and on a closed curve L bounding a simply connected region R then  1 F (t) dt F (z) = for z inside L 2πı L t − z for z outside L

F (z) = 0

The Cauchy integral formula provides a simple means for computing F (z) at interior points when its boundary values are known. We refer to any integral of the form  1 F (t) dt I(z) = 2πi L t − z as a Cauchy integral, regardless of whether F (t) is the boundary value of an analytic function. I(z) deÞnes a function analytic in the complex plane cut along the curve L. When F (t) is the boundary value of a function analytic inside a closed curve L, I(z) is evidently discontinuous across L since I(z) approaches F (z) as z approaches L from the inside but gives zero for an approach from the outside. The theory of Cauchy integrals for both open and closed curves is extensively developed in Muskhelishvili’s texts [72, 73] and is used to solve many practical problems.

12.4.2 Residue Theorem If F (z) is analytic inside and on a closed curve L except at isolated singularities z1 , z2 , . . . , zn where it has Laurent expansions, then  j=n  F (z) dz = 2πı Bj L

j=1

where Bj is the residue of F (z) at z = z j . In the instance where z ı is a pole of order m, the residue can be computed as ' & m−1 1 d m a−1 = [F (z)(z − z ) ] ı lim (m − 1)! dz m−1 z→zı

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12.5 Physical Problems Leading to Analytic Functions Several physical phenomena require solutions involving real valued functions satisfying Laplace’s equation. Since an analytic function has harmonic real and imaginary parts, a harmonic function can often be expressed concisely as the real part of an analytic function. Useful tools such as Taylor series can yield effective computational devices. One of the simplest practical examples involves determining a function u harmonic inside the unit disk |z| ≤ 1 and having boundary values described by a Fourier series. In the following equations, and in subsequent articles, we will often refer to a function deÞned inside and on the unit circle in terms of polar coordinates as u(r, θ) while we may, simultaneously, think of it as a function of the complex variable z = rσ where σ = e iθ . Hence we write the boundary condition for the circular disk as u(1, θ) =

∞ 

cn σ n , σ = eıθ

n=−∞

with c−n = c¯n because u is real. The desired function can be found as u(r, θ) = mboxreal( F (z) ) where F (z) = c0 + 2

∞ 

cn z n

,

|z| ≤ 1

n=1

This solution is useful because the Fast Fourier Transform (FFT) can be employed to generate Fourier coefÞcients for quite general boundary conditions, and the series for F (z) converges rapidly when |z| < 1. This series will be employed below to solve both the problem where boundary values are given (the Dirichlet problem) and where normal derivative values are known on the boundary (the Neumann problem). Several applications where analytic functions occur are mentioned below.

12.5.1 Steady-State Heat Conduction The steady-state temperature distribution in a homogeneous two-dimensional body is harmonic. We can take u = Real[F (z)]. Boundary curves where u = constant lead to conditions F (z) + F (z) = constant in the complex plane. Boundary curves insulated to prevent transverse heat ßow lead to ∂u ∂n = 0, which implies F (z) − F (z) = constant

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12.5.2 Incompressible Inviscid Fluid Flow Some ßow problems for incompressible, nonviscous ßuids involve velocity components obtainable in terms of the Þrst derivative of an analytic function. A complex velocity potential F (z) exists such that u − iv = F  (z) At impermeable boundaries the ßow normal to the boundary must vanish which implies F (z) − F (z) = constant. Furthermore, a uniform ßow Þeld with u = U , v = V is easily described by F (z) = (U − iV )z

12.5.3 Torsion and Flexure of Elastic Beams The distribution of stresses in a cylindrical elastic beam subjected to torsion or bending can be computed using analytic functions [90]. For example, in the torsion problem shear stresses τXZ and τY Z can be sought as z] τXZ − iτY Z = µ ε[f  (z) − i¯ and the condition of zero traction on the lateral faces of the beam is described by f (z) − f (z) = iz z¯ If the function z = ω(ζ) which maps |ζ| ≤ 1 onto the beam cross section is known, then an explicit integral formula solution can be written as 1 f (ζ) = 2π

 |σ|=1

ω(σ)ω(σ)dσ σ−ζ

Consequently, the torsion problem for a beam of simply connected cross section is represented concisely in terms of the function which maps a circular disk onto the cross section.

12.5.4 Plane Elastostatics Analyzing the elastic equilibrium of two-dimensional bodies satisfying conditions of plane stress or plane strain can be reduced to determining two analytic functions. The formulas to Þnd three stress components and two displacement components are more involved than the ones just stated. They will be investigated later when stress concentrations in a plate having a circular or elliptic hole are discussed.

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12.5.5 Electric Field Intensity Electromagnetic Þeld theory is concerned with the Þeld intensity  which is described in terms of the electrostatic potential E [92] such that E = Ex + iEy = −

∂φ ∂φ −i ∂x ∂y

where φ is a harmonic function at all points not occupied by charge. Consequently a complex electrostatic potential Ω(z) exists such that E = −Ω (z) The electromagnetic problem is analogous to inviscid incompressible ßuid ßow problems. We will also Þnd that harmonic functions remain harmonic under the geometry change of a conformal transformation, which will be discussed later. This produces interesting situations where solutions for new problems can sometimes be derived by simple geometry changes.

12.6 Branch Points and Multivalued Behavior Before speciÞc types of maps are examined, we need to consider the concept of branch points. A type of singular point quite different from isolated singularities such as poles arises when a singular point of F (z) cannot be made the interior of a small circle on which F (z) is single valued. Such singularities√are called branch points and the related behavior is typiÞed by functions such as z − z0 and log(z − z0 ). To deÞne p = log(z − z0 ), we accept any value p such that e p produces the value z − z0 . Using polar form we can write (z − z0 ) = |z − z0 |ei(θ+2πk)

where θ = arg(z − z0 )

with k being any integer. Taking p = log |z − z0 | + i(θ + 2πk) yields an inÞnity of values all satisfying e p = z − z0 . Furthermore, if z traverses a counterclockwise circuit around a circle |z − z 0 | = δ, θ increases by 2π and log(z − z0 ) does not return to its initial value. This shows that log(z√− z 0 ) is discontinuous on a path containing z 0 . A similar behavior is exhibited by z − z0 , which changes sign for a circuit about |z − z 0 | = δ. Functions with branch points have the characteristic behavior that the relevant functions are discontinuous on contours enclosing the branch points. Computing √ the function involves selection among a multiplicity of possible values. Hence 4 can equal +2 or −2, and choosing the proper value depends on the functions involved.

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For sake of deÞniteness MATLAB uses what are called principal branch deÞnitions such that

y  √ ≤π z = |z|1/2 eıθ/2 , −π < θ = tan−1 x and log(z) = log |z| + iθ The functions deÞned this way have discontinuities across the negative real axis. Futhermore, log(z) becomes inÞnite at z = 0. Dealing carelessly with multivalued functions can produce strange results. Consider the function p = z2 − 1 which will have discontinuities on lines such that z 2 − 1 = −|h| , where h is a general parameter. Discontinuity trouble occurs when z=±

1 − |h|

Taking 0 ≤ |h| ≤ 1 gives a discontinuity line on the real axis between −1 and +1, and taking |h| > 1 leads to a discontinuity on the imaginary axis. Figure 12.1 illustrates the odd behavior exhibited by sqrt(z.ˆ2-1). The reader can easily verify that using sqrt(z-1).*sqrt(z+1) deÞnes a different function that is continuous in the plane cut along a straight line between −1 and +1. Multivalued functions arise quite naturally in solutions of boundary value problems, and the choices of branch cuts and branch values are usually evident from physical circumstances. For instance, consider a steady-state temperature problem for the region |z| < 1 with boundary conditions requiring u(1, θ) = 1

, 0 < θ < π and

∂u(1, θ) = 0 , π < θ < 2π. ∂r

It can be shown that the desired solution is & ' 1 3 u = real [log(z + 1) − log(z − 1)] + πi 2 where the logarithms must be deÞned so u is continuous inside the unit circle and u equals 1/2 at z = 0. Appropriate deÞnitions result by taking −π < arg(z + 1) ≤ π

,

0 ≤ arg(z − 1) ≤ 2π

MATLAB does not provide this deÞnition intrinsically; so, the user must handle each problem individually when branch points arise.

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Discontinuous Surface for imag( sqrt( z2 − 1 ) )

3 imag( sqrt( z 2−1 ) )

2 1 0 −1 −2 −3 2 1.5

2

1 0.5

1 0

0

−0.5 −1 imaginary axis

−1

−1.5 −2

−2

real axis

Figure 12.1: Discontinuous Surface for imag (sqrt (z 2 − 1)1/2 )

12.7 Conformal Mapping and Harmonic Functions A transformation of the form x = x(ξ, η) , y = y(ξ, η) is said to be conformal if the angle between intersecting curves in the (ξ, η) plane remains the same for corresponding mapped curves in the (x, y) plane. Consider the transformation implied by z = ω(ζ) where ω is an analytic function of ζ. Since dz = ω  (ζ) dζ it follows that |dz| = |ω  (ζ)| |dζ|

and

arg(dz) = arg( ω  (ζ) ) + arg(dζ)

This implies that the element of length |dζ| is stretched by a factor of |ω  (ζ)| and the line element dζ is rotated by an angle arg[ω  (ζ)]. The transformation is conformal at all points where ω  (ζ) exists and is nonzero. Much of the interest in conformal mapping results from the fact that harmonic functions remain harmonic under a conformal transformation. To see why this is true, examine Laplace’s equation written in the form ∇2xy u = 4

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∂2u =0 ∂z∂ z¯

For a conformal map we have z = ω(ζ)

,

z¯ = ω(ζ)

1 ∂u ∂u =  ∂z ω (ζ) ∂ζ

,

∂u 1 ∂u =  ∂ z¯ ω (ζ) ∂ ζ¯

Since z depends only on ζ and z¯ depends only on ζ¯ we Þnd that ∇2xy u = 4 It follows that

1 ∂2u ∇2 u =  ¯   |ω (ζ)|2 ξη ω (ζ)ω (ζ) ∂ζ∂ ζ 1

∂2u ∂2u + 2 =0 ∂x2 ∂y

implies

∂2u ∂2u + 2 =0 ∂ξ 2 ∂η

wherever ω  (ζ) = 0. The transformed differential equation in the new variables is identical to that of the original differential equation. Hence, when u(x, y) is a harmonic function of (x, y), then u(x(ξ, η), y(ξ, η)) is a harmonic function of (ξ, η), provided ω(ζ) is an analytic function. This is a remarkable and highly useful property. Normally, changing the independent variables in a differential equation changes the form of the equation greatly. For instance, with the polar coordinate transformation x = r cos(θ), y = r sin(θ) the Laplace equation becomes ∇2 u =

1 ∂2u ∂ 2 u 1 ∂u + 2 2 =0 + 2 ∂r r ∂r r ∂θ

The appearance of this equation is very different from the Cartesian form because x+iy is not an analytic function of r+iθ. On the other hand, using the transformation z = log(ζ) = log(|ζ|) + i arg(ζ) gives ¯ 2 u ∇2xy u = (ζ ζ)∇ ξη and ∇2xy u = 0 implies ∇2ξη u = 0 at points other than ζ = 0 or ζ = ∞. Because solutions to Laplace’s equation are important in physical applications, and such functions remain harmonic under a conformal map, an analogy between problems in two regions often can be useful. This is particularly attractive for problems where the harmonic function has constant values or zero normal gradient on critical boundaries. An instance pertaining to inviscid ßuid ßow about an elliptic cylinder will be used later to illustrate the harmonic function analogy. In the subsequent sections we discuss several transformations and their relevant geometrical interpretation.

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12.8 Mapping onto the Exterior or the Interior of an Ellipse We will examine in some detail the transformation     a+b a−b z= ζ+ ζ −1 = R(ζ + mζ −1 ) 2 2

,

ζ ≥1

where R = (a + b)/2 and m = (a − b)/(a + b). The derivative z  (ζ) = R(1 − mζ −2 ) √ = 0 or ζ = ± m. For sake of discussion, we becomes nonconformal when z  (ζ) √ temporarily assume a ≥ b to make m real rather than purely imaginary. A circle ζ = ρ0 eıθ transforms into −1 x + ıy = R(ρ0 + mρ−1 0 ) cos(θ) + iR(ρ0 − mρ0 ) sin(θ)

yielding an ellipse. When ρ 0 = 1 we get x = a cos(θ), y = b sin(θ). This mapping function is useful in problems such as inviscid ßow around an elliptic cylinder or stress concentration around an elliptic hole in a plate. Furthermore, the mapping function is easy to invert by solving a quadratic equation to give z + (z − α)(z + α) ζ= , α = a2 − b 2 a+b The radical should be deÞned to have a branch cut on the x-axis from −α to α and to behave like +z for large |z|. Computing the radical in MATLAB as sqrt(z-alpha).*sqrt(z+alpha) works Þne when α is real because MATLAB uses −π < arg(z ± α) ≤ π and the sign change discontinuities experienced by both factors on the negative real axis cancel to make the product of radicals √ continuous. However, when a < b the branch points occur at ±z 0 where z0 = i b2 − a2 , and a branch cut is needed along the imaginary axis. We can give a satisfactory deÞnition by requiring −

π 3π < arg(z ± z0 ) ≤ 2 2

The function elipinvr provided below handles general a and b. Before leaving the problem of ellipse mapping we mention the fact that mapping the interior of a circle onto the interior of an ellipse is rather complicated but can be formulated by use of elliptic functions [75]. However, a simple solution to compute boundary point correspondence between points on the circle and points on the ellipse

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appears in [52]. This can be used to obtain mapping functions in rational form which are quite accurate. The function elipdplt produces the mapping. Results showing how a polar coordinate grid in the ζ-plane maps onto a two to one ellipse appears in Figure 12.2. In these examples and other similar ones, grid networks in polar coordinates always use constant radial increments and constant angular increments. Only the region corresponding to 0.3 ≤ |ζ| ≤ 1 and 0 ≤ arg(ζ) ≤ π2 is shown. Note that the distortion of line elements at different points of the grid is surprisingly large. This implies that the stretching effect, depending on |ω  (ζ)| ,varies more than might at Þrst be expected. Often it is desirable to see how a rectangular or polar coordinate grid distorts under a mapping transformation. This is accomplished by taking the point arrays and simultaneously plotting rows against rows and columns against columns as computed by the following function gridview which works for general input arrays x, y. If the input data are vectors instead of arrays, then the routine draws a single curve instead of a surface. When gridview is executed with no input, it generates the plot in Figure 12.3 which shows how a polar coordinate grid in the ζ-plane maps under the transformation   m z=R ζ+ ζ The new grid consists of a system of confocal ellipses orthogonally intersecting a system of hyperbolas.

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Mapping abs(ZETA)=1 onto (x/a).^2+(y/b).^2 >= 1 % % a - semi-diameter on x-axis % b - semi-diameter on y-axis % z - array of complex values % % zeta - array of complex values for the % inverse mapping function % % User m functions called: none

18:

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19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29:

z0=sqrt(a^2-b^2); ab=a+b; if a==b zeta=z/a; elseif a>b % branch cut along the real axis zeta=(z+sqrt(z-z0).*sqrt(z+z0))/ab; else % branch cut along the imaginary axis ap=angle(z+z0); ap=ap+2*pi*(ap