Handbook of mathematics for engineers and scienteists part 63 pot

If the function w = f z is single-valued on D and the images of distinct points of D are distinct, then the mapping determined by this function is said to be schlicht.. The notions of bo

of D with its boundary is called a closed domain and is denoted by D The positive sense of

the boundary is defined to be the sense for which the domain lies to the left of the boundary The boundary of a domain can consist of finitely many closed curves, segments, and points; the curves and cuts are assumed to be piecewise smooth

The simplest examples of domains are neighborhoods of points on the complex plane

A neighborhood of a point a on the complex plane is understood as the set of points z such

that|z – a| < R, i.e., the interior of the disk of radius R >0centered at the point a The

extended complex plane is obtained by augmenting the complex plane with the fictitious point at infinity A neighborhood of the point at infinity is understood as the set of points z

such that|z|> R (including the point at infinity itself).

If to each point z of a domain D there corresponds a point w (resp., a set of points w), then one says that there is a single-valued (resp., multivalued) function w = f (z) defined

on the domain D If we set z = x + iy and w = u + iv, then defining a function w = f (z)

of the complex variable is equivalent to defining two functions Re f = u = u(x, y) and

Im f = v = v(x, y) of two real variables If the function w = f (z) is single-valued on D and the images of distinct points of D are distinct, then the mapping determined by this function

is said to be schlicht The notions of boundedness, limit, and continuity for single-valued

functions of the complex variable do not differ from the corresponding notions for functions

of two real variables

10.1.2-2 Differentiability and analyticity

Let a single-valued function w = f (z) be defined in a neighborhood of a point z If there

exists a limit

lim

h→0

f (z + h) – f (z)

then the function w = f (z) is said to be differentiable at the point z and f z  (z) is called its

derivative at the point z.

Cauchy–Riemann conditions If the functions u(x, y) = Re f (z) and v(x, y) = Im f (z)

are differentiable at a point (x, y), then the Cauchy–Riemann conditions

∂u

∂x = ∂v

∂y = –∂v

are necessary and sufficient for the function w = f (z) to be differentiable at the point z = x+iy.

If the function w = f (z) is differentiable, then

w 

z = u x + iv x = v y – iu y = u x + iu y = v y + iv x, (10.1.2.3)

where the subscripts x and y indicate the corresponding partial derivatives.

Remark The Cauchy–Riemann conditions are sometimes also called the d’Alembert–Euler conditions.

The rules for algebraic operations on the derivatives and for calculating the derivative

of the composite function and the inverse function (if it exists) have exactly the same form

as in the real case:

1 

αf1(z) βf2(z)

z = α[f1(z)]  z β [f2(z)]  z , where α and β are arbitrary complex constants

2 

f1(z)f2(z)

z = [f1(z)]  z f2(z) + f1(z)[f2(z)]  z. 3

*f1(z)

f2(z)

+

z =

[f1(z)]  z f2(z) – f1(z)[f2(z)]  z

f2

0)

4 If a function w = f (z) is differentiable at a point z and a function W = F (w) is differentiable at the point w = f (z), then the composite function W = F (f (z)) is differentiable at the point z and W z  = [F (f (z))]  z = F f  (f )f z  (z).

5 If a function w =f (z) is differentiable at a point z and the inverse function z =g(w)≡f–1(w)

exists and is differentiable at the point w and satisfies [f–1(w)]  w ≠ 0, then

f 

[f– 1(w)] 

w.

A single-valued function differentiable in some neighborhood of a point z0is said to be

analytic (regular, holomorphic) at this point.

A function w = f (z) is analytic at a point z0 if and only if it can be represented by a power series

f (z) =

∞



k=0

converging in some neighborhood of z0

A function analytic at each point of the domain D is said to be analytic in D.

A function w = f (z) is said to be analytic at the point at infinity if the function

F (z) = f (1/z ) is analytic at the point z =0 In this case, f z (∞) = (–z2F 

z)z=0by definition.

A function w = f (z) is analytic at the point at infinity if and only if this function can be

represented by a power series

f (z) =

∞



k=0

converging for sufficiently large|z|

If a function w = f (z) is analytic at a point z0and f z  (z0)≠ 0, then f (z) has an analytic inverse function z(w) defined in a neighborhood of the point w0 = f (z0) If a function

w = f (z) is analytic at a point z0 and the function W = F (w) is analytic at the point

w0 = f (z0), then the composite function W = F [f (z)] is analytic at the point z0 If a

function is analytic in a domain D and continuous in D, then its value at any interior point

of the domain is uniquely determined by its values on the boundary of the domain The analyticity of a function at a point implies the existence and analyticity of its derivatives of arbitrary order at this point

MAXIMUM MODULUS PRINCIPLE If a function w = f (z) that is not identically constant

is analytic in a domain D and continuous in D, then its modulus cannot attain a maximum

at an interior point of D.

LIOUVILLE’S THEOREM If a function w = f (z) is analytic and bounded in the entire

complex plane, then it is constant

Remark The Liouville theorem can be stated in the following form:

if a function w = f (z) is analytic in the extended complex plane, then it is constant.

Geometric meaning of the absolute value of the derivative Suppose that a function

w = f (z) is analytic at a point z0 and f z  (z0) ≠ 0 Then the value|f 

z (z0)| determines the

dilatation (similarity) coefficient at the point z0 under the mapping w = f (z) The value

|f 

z (z0)|is called the dilatation ratio if|f 

z (z0)|>1and the contraction ratio if|f 

z (z0)|<1

Geometric meaning of the argument of the derivative The argument of the derivative

f 

z (z0) is equal to the angle by which the tangent at the point z0 to any curve passing

through z0 should be rotated to give the tangent to the image of the curve at the point

w0= f (z0) For ϕ = arg f z  (z) >0, the rotation is anticlockwise, and for ϕ = arg f z  (z) <0, the rotation is clockwise

Single-valued functions, as well as single-valued branches of multi-valued functions, are analytic everywhere on the domains where they are defined It follows from (10.1.2.2)

that the real and imaginary parts u(x, y) and v(x, y) of a function analytic in a domain are

harmonic in this domain, i.e., satisfy the Laplace equation

in this domain

Remark. If u(x, y) and v(x, y) are two arbitrary harmonic functions, then the function f (z) = u(x, y) + iv(x, y) is not necessarily analytic, since for the analyticity of f (z) the functions u(x, y) and v(x, y) must satisfy

the Cauchy–Riemann conditions.

Example 1 The function w = z2is analytic.

Indeed, since z = x + iy, we have w = (x + iy)2= x2– y2+ i2xy, u(x, y) = x2– y2, and v(x, y) =2xy The Cauchy–Riemann conditions

u x = v y= 2x, u y = v x= –2y

are satisfied at all points of the complex plane, and the function w = z2is analytic.

Example 2 The function w = ¯z is not analytic.

Indeed, since z = x + iy, we have w = x – iy, u(x, y) = x, v(x, y) = –y The Cauchy–Riemann conditions

are not satisfied,

u x= 1 ≠ –1= v y, u y = v x= 0,

and the function w = ¯z is not analytic.

10.1.2-3 Elementary functions

1◦ The functions w = z n and w = √ n

z for positive integer n are defined in Paragraph 10.1.1-2.

The function

is single-valued It is schlicht in the sectors2πk/n < ϕ <2π (k +1)/n, k =0,1,2, , each

of which is transformed by the mapping w = z n into the plane w with a cut on the positive

semiaxis

The function

w= √ n

is an n-valued function for z≠ 0, and its value is determined by the value of the argument

chosen for the point z If a closed curve C does not surround the point z =0, then, as the

point z goes around the entire curve C, the point w = √ n

zfor a chosen value of the root also moves along a closed curve and returns to the initial value of the argument But if

the curve C surrounds the origin, then, as the point z goes around the entire curve C in the positive sense, the argument of z increases by2π and the corresponding point w = √ n

z does not return to the initial position It will return there only after the point z goes n times around the entire curve C If a domain D does not contain a closed curve surrounding the point z =0, then one can singe out n continuous single-valued functions, each of which takes only one of the values w = √ n

z ; these functions are called the branches of the multi-valued function w = √ n

z One cannot single n separate branches of the function w = √ n

z in any

neighborhood of the point z =0; accordingly, the point z =0is called a branch point of this

function

2◦ The Zhukovskii function

w= 1 2



z+ 1

z



(10.1.2.9)

is defined and single-valued for all z ≠ 0; it is schlicht in any domain that does not

simultaneously contain any points z1and z2such that z1z2=1

3◦ The exponential function w = e zis defined by the formula

w = e z = e x+iy = e x (cos y + i sin y). (10.1.2.10)

The function w = e z is analytic everywhere For the exponential function, the usual differentiation rule is preserved:

(e z) z = e z

The basic property of the exponential function (addition theorem) is also preserved:

e z1 e z2 = e z1+z2. For x =0and y = ϕ, the definition of the exponential function implies the Euler formula

e iϕ = cos ϕ + i sin ϕ, which permits one to write any complex number with modulus r and argument ϕ in the exponential form

z = r(cos ϕ + i sin ϕ) = re iϕ (10.1.2.11) The exponential function is 2π -periodic, and the mapping w = e z is schlicht in the strip0 ≤y<2π

4◦ The logarithm is defined as the inverse of the exponential function: if e w = z, then

This function is defined for z≠ 0 The logarithm satisfies the following relations:

Ln z1+ Ln z2= Ln(z1z2), Ln z1– Ln z2= Ln z1

z2,

Ln(z n ) = n Ln z, Ln√ n

z= 1

n Ln z.

The exponential form of complex numbers readily shows that the logarithm is infinite-valued:

Ln z = ln|z|+ i Arg z = ln|z|+ i arg z +2πki, k=0, 1, 2, (10.1.2.13)

The value ln z = ln|z|+ i arg z is taken to be the principal value of this function Just as with the function w = √ n

z , we see that if the point z =0is surrounded by a closed curve C, then the point w = Ln z does not return to its initial position after z goes around C in the positive sense, since the argument of w increases by2πi Thus if a domain D does not contain a closed curve surrounding the point z =0, then in D one can single out infinitely many continuous and single-valued branches of the multi-valued function w = Ln z; the

differences between the values of these branches at each point of the domain have the form

2πki , where k is an integer This cannot be done in an arbitrary neighborhood of the point

z=0, and this point is called a branch point of the logarithm.

5◦ Trigonometric functions are defined in terms of the exponential function as follows:

cos z = e iz + e

–iz

e iz – e–iz

2i ,

tan z = sin z

cos z = –i

e iz – e–iz

e iz + e–iz, cot z =

cos z sin z = i

e iz + e–iz

e iz – e–iz.

(10.1.2.14)

Properties of the functions cos z and sin z:

1 They are analytic for any z.

2 The usual differentiation rules are valid:

(sin z)  z = cos z, (cos z)  z = sin z.

3 They are periodic with real period T =2π

4 sin z is an odd function, and cos z is an even function.

5 In the complex domain, they are unbounded

6 The usual trigonometric relations hold:

cos2z+ sin2z=1, cos2z= cos2z– sin2z, etc

The function tan z is analytic everywhere except for the points

z k= π2 + kπ, k=0, 1, 2, ,

and the function cot z is analytic everywhere except for the points

z k = kπ, k=0, 1, 2, The functions tan z and cot z are periodic with real period T = π.

6◦ Hyperbolic functions are defined by the formulas

cosh z = e z + e

–z

e z – e–z

2 ,

tanh z = sinh z

cosh z =

e z – e–z

e z + e–z, coth z =

cosh z sinh z =

e z + e–z

e z – e–z.

(10.1.2.15)

For real values of the argument, each of these functions coincides with the corresponding real function Hyperbolic and trigonometric functions are related by the formulas

cosh z = cos iz, sinh z = –i sin iz, tanh z = –i tan iz, coth z = i cot iz.

7◦ Inverse trigonometric and hyperbolic functions are expressed via the logarithm and

hence are infinite-valued:

Arccos z = –i Ln(z + √

z2–1), Arcsin z = –i Ln(iz + √

1– z2),

Arctan z = – i

2Ln

1+ iz

i

2 Ln

z + i

z – i, arccosh z = Ln(z + √

z2–1), arcsinh z = Ln(z + √

z2–1),

arctanh z = 1

2 Ln

1+ z

1

2 Ln

z+1

z–1.

(10.1.2.16)

The principal value of each of these functions is obtained by choosing the principal value of the corresponding logarithmic function

8◦ The power w = z γis defined by the relation

z γ = e γ Ln z, (10.1.2.17)

where γ = α + iβ is an arbitrary complex number Substituting z = re iϕinto (10.1.2.17),

we obtain

z γ = e α ln r–β(ϕ+2kπ) e iα(ϕ+2kπ)+iβ ln r, k=0, 1, 2, (10.1.2.18)

It follows from relation (10.1.2.18) that the function w = z γ has infinitely many values

for β ≠ 0

9◦ The general exponential function is defined by the formula

w = γ z = e z Ln γ = e z ln|γ|e zi Arg γ, (10.1.2.19)

where γ = α + iβ is an arbitrary nonzero complex number The function (10.1.2.19) is a

set of separate mutually independent single-valued functions that differ from one another

by the factors e2kπiz, k =0, 1, 2,

Example 3 Let us calculate the values of some elementary functions at specific points:

1 cos 2i = 12(e2ii + e– 2ii) = 12(e2+ e– 2 ) = cosh 2 ≈ 3.7622.

2 ln(–2) = ln 2+ iπ, since| – 2| = 2and the principal value of the argument is equal to π.

3 Ln(–2) is calculated by formula (10.1.2.13):

Ln(–2) = ln 2+ iπ +2πki = ln 2 + (1 + 2k)iπ (k =0, 1, 2, ).

4 i i = e i Ln i = e i(iπ/2+ 2πk) = e–π/2 – 2πk (k =0, 1, 2, ).

The main elementary functions w = f (z) = u(x, y) + iv(x, y) of the complex variable

z = x + iy are given in Table 10.1.

10.1.2-4 Integration of function of complex variable

Suppose that an oriented curve C connecting points z = a and z = b is given on the complex plane and a function w = f (z) of the complex variable is defined on the curve We divide the curve C into n parts, a = z0, z1, , z n–1 , z n = b, arbitrarily choose ξ k[z k , z k+1], and compose the integral sum

n–1



k=0

f (ξ k )(z k+1 – z k)

If there exists a limit of this sum as max|z k+1 – z k|→0, independent of the construction of

the partition and the choice of points ξ k , then this limit is called the integral of the function

w = f (z) over the curve C and is denoted by



Properties of the integral of a function of a complex variable:

1 If α, β are arbitrary constants, then7

C [αf (z) + βg(z)] dz = α

7

C f (z) dz + β

7

C g (z) dz.

2 If 2C is the same curve as C but with the opposite sense, then7

2

C f (z) dz = –

7

C f (z) dz.

3 If C = C1∪ · · · ∪ C n, then7

C f (z) dz =

7

C1 f (z) dz + +

7

C n f (z) dz.

4 If|f (z)| ≤M at all points of the curve C, then the following estimate of the absolute

value of the integral holds: 7

C f (z) dz≤M l , where l is the length of the curve C.

If C is a piecewise smooth curve and f (z) is bounded and piecewise continuous, then the integral (10.1.2.20) exists If z = x + iy and w = u(x, y) + iv(x, y), then the computation

of the integral (10.1.2.20) is reduced to finding two ordinary curvilinear integrals:



C f (z) dz =



C u (x, y) dx – v(x, y) dy + i



C v (x, y) dx + u(x, y) dy. (10.1.2.21) Remark Formula (10.1.2.21) can be rewritten in a form convenient for memorizing:



C

f(z) dz =



C

(u + iv)(dx + i dy).

TABLE 10.1

Main elementary functions w = f (z) = u(x, y) + iv(x, y) of complex variable z = x + iy

No.

Complex

function

w = f (z)

Algebraic form

f(z) = u(x, y) + iv(x, y) Zeros of nth order Singularities

is a first-order pole

2 z2 x2– y2+ i2xy z= 0, n = 2 z=∞

is a second-order pole

3

1

z– (x0+iy0)

(x0, y0 are

real numbers)

x–x0 (x–x0)2+ (y–y0)2 + i

–(y–y0)

(x–x0)2+ (y–y0)2 z=∞, n =1 z = x0+ iy0

is a first-order pole

z2

x2– y2 (x2+ y2)2 + i

–2xy

(x2+ y2)2 z=∞, n =2 z=0

is a second-order pole

5 √ z x+

x2+y2

2

1/2

+i  –x+x2+y2

2

1/2 z=0is a branch

point

z= 0 is a first-order branch point

z=∞ is a first-order

branch point

singular point

7 Ln z ln|z|+ i(arg z +2kπ),

k= 0, 1, 2,

z= 1, n = 1 (for the branch corresponding

to k =0)

Logarithmic branch points

for z =0, z =∞

8 sin z sin x cosh y + i cos x sinh y z = πk, n =1

(k =0, 1, 2, )

z=∞ is an essential

singular point

9 cos z cos x cosh y + i(– sin x sinh y) z= 12π + πk, n =1

(k =0, 1, 2, )

z=∞ is an essential

singular point

cos 2x + cosh 2y + i

sinh 2y cos 2x + cosh 2y (k = z = πk, n =0, 1, 2, )1

z= 12π + πk (k =0, 1, 2, ) are first-order poles

If the curve C is given by the parametric equations x = x(t), y = y(t) (t1≤t≤t2), then



C f (z) dz =

 t2

t1 f (z(t))z



t (t) dt, (10.1.2.22)

where z = z(t) = x(t) + iy(t) is the complex parametric equation of the curve C.

If f (z) is an analytic function in a simply connected domain D containing the points

z = a and z = b, then the Newton–Leibniz formula holds:

 b

a f (z) dz = F (b) – F (a), (10.1.2.23)

where F (z) is a primitive of the function f (z), i.e., F z  (z) = f (z) in the domain D.

If f (z) is an analytic function in a simply connected domain D containing the points

z = a and z = b, then the Newton–Leibniz formula holds:...

z= 1, n = (for the branch corresponding

to k =0)

Logarithmic branch points

for z =0,... (z) dz = F (b) – F (a), (10.1.2.23)

where F (z) is a primitive of the function f (z), i.e., F z  (z) = f (z) in the domain D.