Handbook of mathematics for engineers and scienteists part 67 ppt

S., Complex Variables: Introduction and Applications Cambridge Texts in Applied Mathematics, 2nd Edition, Cambridge University Press, Cambridge, 2003.. and Roger Gay, R., Complex Variabl

430 FUNCTIONS OFCOMPLEXVARIABLE

10.2.2-3 Cauchy-type integral

Suppose that C is an arbitrary curve without cusps, not necessarily closed Let an arbitrary function f (ξ), which is assumed to be finite and integrable, be given on this curve.

The integral

F (z) = 1

2πi



C

f (ξ) dξ

is called a Cauchy-type integral.

The Cauchy-type integral is a function analytic at any point z that does not lie on C.

If the curve C divides the plane into several domains, then, in general, the Cauchy-type

integral defines different analytic functions in these domains

One says that the function f (ξ) satisfies the H¨older condition with exponent μ≤ 1at a

point ξ = ξ0of the contour C if there exists a constant M such that the inequality

|f (ξ) – f (ξ0)| ≤M|ξ – ξ0|μ (0< μ≤ 1) (10.2.2.14)

holds for all points ξ C sufficiently close to ξ0 The H¨older condition means that the

increment of the function is an infinitesimal of order at least μ with respect to the increment

of the argument

The principal value of the integral is defined as the limit

lim

r→0



C–c

f (ξ) dξ

ξ – ξ0 =



C

f (ξ) dξ

where c is the segment of the curve C between the points of intersection of C with the circle

|z – ξ0|= r.

The singular integral in the sense of the Cauchy principal value is defined as the integral

given by the formula



C–c

f (ξ) dξ

ξ – ξ0 =



C

f (ξ) – f (ξ0)

ξ – ξ0 dξ + f (ξ0) ln

b – ξ0

a – ξ0 + iπf (ξ0) + O(r), (10.2.2.16)

where a and b are the endpoints of C and O(r) →0as r →0

THEOREM If the function f (ξ) satisfies the H¨older condition with exponent μ≤ 1at a

point ξ0which is a regular (nonsingular) point of the contour C and does not coincide with

its endpoints, then the Cauchy-type integral exists at this point as a singular integral and its principal value can be expressed in terms of the usual integral by the formula

F (ξ0) = 1

2πi



C

f (ξ) dξ

ξ – ξ0 =

1

2πi



C

f (ξ) – f (ξ0)

ξ – ξ0 dξ+

f (ξ0)

2 +

f (ξ0)

2πi ln b – ξ0

a – ξ0 (10.2.2.17)

If the curve C is closed, then a = b and formula (10.2.2.17) becomes

F (ξ0) = 1

2πi



C

f (ξ) dξ

ξ – ξ0 =

1

2πi



C

f (ξ) – f (ξ0)

ξ – ξ0 dξ+

f (ξ0)

2 . (10.2.2.18)

Suppose that the function f (ξ) satisfies the H¨older condition with exponent μ≤ 1at the

point ξ = ξ0and the point z tends to ξ0so that the ratio of h =|z – ξ0|to d (dh is the shortest distance from z to the points of C) remains bounded Then

lim

z→ξ0



C

f (ξ) – f (ξ0)

ξ – z dξ=



C

f (ξ) – f (ξ0)

SOKHOTSKII’S THEOREM Suppose that ξ0 is a regular (nonsingular) point of the

con-tour C and does not coincide with its endpoints, the function f (ξ) satisfies the H¨older condition with exponent μ ≤ 1 at this point, and z → ξ0 so that the ratio h/d remains bounded Then the Cauchy-type integral has limit values F+(ξ0)and F–(ξ0)to which this

integral tends as z → ξ0from the left and, respectively, from the right of C, and

F+(ξ0) = F (ξ0) + 1

2f (ξ0), F–(ξ0) = F (ξ0) –

1

2f (ξ0), (10.2.2.20)

where F (ξ0)is the singular integral (10.2.2.18)

The Cauchy-type integral experiences a jump when passing through the integration

contour C at the point ξ0:

F+(ξ0) – F–(ξ0) = f (ξ0) (10.2.2.21) The condition

at each point of C is necessary and sufficient for a Cauchy-type integral to be the Cauchy

integral

THEOREM If a function f (ξ) satisfies the H¨older condition with exponent μ≤ 1at each

point of a closed contour C, then, for its values to be the boundary values of a function analytic in the interior of C, it is necessary and sufficient that



C ξ

n f (ξ) dξ =0 (n =0,1,2, ). (10.2.2.23)

THEOREM If a function f (ξ) satisfies the H¨older condition with exponent μ≤ 1at each

point of a closed contour C, then, for the values of f (ξ) to be the boundary values of a function analytic in the interior of C, it is necessary and sufficient that

1

2πi



C

f (ξ) dξ

for all points z lying in the exterior of C.

THEOREM If a function f (ξ) satisfies the H¨older condition with exponent μ≤ 1at each

point of a closed contour C, then, for the values of f (ξ) to be the boundary values of a function analytic in the exterior of C, it is necessary and sufficient that

1

2πi



C

f (ξ) dξ

for all points z lying in the interior of C.

THEOREM For the values of a function f (ξ) satisfying the H¨older condition with ex-ponent μ≤ 1to be the boundary values of a function analytic (a) in the interior of the disk

|z| < 1 or (b) in the exterior of this disk, it is necessary and sufficient that the following respective conditions hold:

for all z in the interior of C,

1

2πi



C

f (ξ) dξ

ξ – z = f (0), (10.2.2.26)

for all z in the exterior of C

1

2πi



C

f(ξ) dξ

432 FUNCTIONS OFCOMPLEXVARIABLE

Example (the first main problem of elasticity).

Let D be the unit disk Find the elastic equilibrium for given external stresses F n = X n + iY non the unit

circle C, where X n and Y nare the components of a surface force vector.

The problem is to find functions ϕ and ψ satisfying the boundary condition

ϕ (ξ) + ξϕ  ξ (ξ) + ψ(ξ) = f (ξ), where f (ξ) = i7ξ

ξ0F n ds is a function given on C.

To be definite, we set

ψ( 0) = Im ϕ  ξ( 0 ) = 0

By formula (10.2.2.26), the relation

1

2πi



C

ψ (ξ) dξ

ξ – z =

1

2πi



C

f (ξ) dξ

ξ – z –

1

2πi



C

ϕ (ξ) dξ

ξ – z –

1

2πi



C

ξϕ  ξ (ξ)

ξ – z dξ=0

holds for all |z| < 1 Since the function ϕ(z) is analytic in the disk|z| < 1 , we can use the Cauchy formula and rewrite this relation as

ϕ (z) + 1

2πi



C

ξϕ  ξ (ξ)

ξ – z dξ=

1

2πi



C

f (ξ) dξ

ξ – z . Thus we obtain an equation for the function ϕ(z) Omitting the details, we write out the definitive result:

ϕ (z) = 1

2πi



C

f (ξ) dξ

ξ – z –

z

4πi



C

f (ξ) dξ

ξ2 .

To find the function ψ(z), we pass from the boundary condition ϕ(ξ) + ξϕ  ξ (ξ) + ψ(ξ) = f (ξ) to the complex conjugate condition and solve it for ψ(z) Thus we obtain

ψ (ξ) = f (ξ) – ϕ(ξ) – ξϕ  ξ (ξ).

We calculate the Cauchy-type integral of the expressions in both sides, which is reduced to the Cauchy integral

in either case, and obtain

ψ (z) = 1

2πi



C

f (ξ) dξ

ξ – z +

1

4πiz



C

f (ξ) dξ

ξ2 –ϕ



ξ (ξ)

z .

10.2.2-4 Hilbert–Privalov boundary value problem

Privalov boundary value problem Given two complex functions a(ξ)≠ 0and b(ξ) satisfying the H¨older condition with exponent μ≤ 1on a closed curve C, find a function f–(z) analytic

in the exterior of C including the point at infinity z = ∞ and a function f+(z) analytic in the interior of C such that the boundary values f–(ξ) and f+(ξ) of these functions on C exist

and satisfy the relation

f–(ξ) = a(ξ)f+(ξ) + b(ξ). (10.2.2.28)

If b(ξ) =0, i.e., if the boundary relation has the form

then the Privalov boundary value problem is called the Hilbert boundary value problem The index (winding number) of a function a(ξ) is defined to be the integer equal to the net increment of its argument along the closed curve C, divided by2π:

1

2πΔC arg a(ξ) = 1

2πi



C d ln a(ξ). (10.2.2.30) GAKHOV’S FIRST THEOREM The Hilbert problem

f–(ξ) = a(ξ)f+(ξ) has a family of solutions depending on n +1 arbitrary constants if the index n of the boundary function a(ξ) is not positive If the index n is positive, then the problem does not

have solutions analytic in the corresponding domains

The solutions of the Hilbert problem can be written as

f–(ξ) =



a0+ a1

z +· · · + a n

z n



exp[–F1–(z)],

f+(ξ) = (a0z n + a1z n–1+· · · + a n ) exp[–F1+(z)],

(10.2.2.31)

where

F1(z) = 1

2πi



C

ln[ξ n a(ξ)]

ξ – z dξ.

The constants a0, , a n in formula (10.2.2.31) are arbitrary, and a0is determined by the

choice of the value f–(∞).

GAKHOV’S SECOND THEOREM The Privalov problem

f–(ξ) = a(ξ)f+(ξ) + b(ξ) has a family of solutions depending on n +1 arbitrary constants if the index n of the boundary function a(ξ) is not positive If the index n of the function a(ξ) is positive, then the problem is solvable only if the function b(ξ) satisfies the condition



C

b(ξ) exp[–F1–(ξ)]

ξ k+1 dξ=0 (k =1,2, , n). (10.2.2.32)

The solutions of the Privalov problem can be written as

f–(ξ) =*

a0+ a z1 + + a z n n + F2–(z)

+

exp[–F1–(z)],

f+(ξ) =

a0z n + a1z n–1 + + a

n + z n F2+(z)

exp[–F1+(z)],

(10.2.2.33)

where a0, , a n are arbitrary constants and F2(z) is determined by the formula

F2(z) = – 1

2πi



C

b(ξ) exp[–F1(ξ)]

References for Chapter 10

Ablowitz, M J and Fokas, A S., Complex Variables: Introduction and Applications (Cambridge Texts in

Applied Mathematics), 2nd Edition, Cambridge University Press, Cambridge, 2003.

Berenstein, C A and Roger Gay, R., Complex Variables: An Introduction (Graduate Texts in Mathematics),

Springer, New York, 1997.

Bieberbach, L., Conformal Mapping, American Mathematical Society, Providence, Rhode Island, 2000 Bronshtein, I N., Semendyayev, K A., Musiol, G., and M ¨uhlig, H., Handbook of Mathematics, 4th Edition,

Springer, New York, 2004.

Brown, J W and Churchill, R V., Complex Variables and Applications, 7th Edition, McGraw-Hill, New

York, 2003.

Caratheodory, C., Conformal Representation, Dover Publications, New York, 1998.

Carrier, G F., Krock, M., and Pearson, C E., Functions of a Complex Variable: Theory and Technique

(Classics in Applied Mathematics), Society for Industrial & Applied Mathematics, University City Science Center, Philadelphia, 2005.

Cartan, H., Elementary Theory of Analytic Functions of One or Several Complex Variables, Dover Publications,

New York, 1995.

Conway, J B., Functions of One Complex Variable I (Graduate Texts in Mathematics), 2nd Edition, Springer,

New York, 1995.

434 FUNCTIONS OFCOMPLEXVARIABLE

Conway, J B., Functions of One Complex Variable II (Graduate Texts in Mathematics), 2nd Edition, Springer,

New York, 1996.

Dettman, J W., Applied Complex Variables (Mathematics Series), Dover Publications, New York, 1984.

England, A H., Complex Variable Methods in Elasticity, Dover Edition, Dover Publications, New York, 2003 Fisher, S D., Complex Variables (Dover Books on Mathematics), 2nd Edition, Dover Publications, New York,

1999.

Flanigan, F J., Complex Variables, Dover Ed Edition, Dover Publications, New York, 1983.

Greene, R E and Krantz, S G., Function Theory of One Complex Variable (Graduate Studies in Mathematics),

Vol 40, 2nd Edition, American Mathematical Society, Providence, Rhode Island, 2002.

Ivanov, V I and Trubetskov, M K., Handbook of Conformal Mapping with Computer-Aided Visualization,

CRC Press, Boca Raton, 1995.

Korn, G A and Korn, T M., Mathematical Handbook for Scientists and Engineers: Definitions, Theorems,

and Formulas for Reference and Review, Dover Edition, Dover Publications, New York, 2000.

Krantz, S G., Handbook of Complex Variables, Birkh¨auser, Boston, 1999.

Lang, S., Complex Analysis (Graduate Texts in Mathematics), 4th Edition, Springer, New York, 2003 Lavrentiev, M A and Shabat, V B., Methods of the Theory of Functions of a Complex Variable, 5th Edition

[in Russian], Nauka Publishers, Moscow, 1987.

LePage, W R., Complex Variables and the Laplace Transform for Engineers, Dover Publications, New York,

1980.

Markushevich, A I and Silverman, R A (Editor), Theory of Functions of a Complex Variable, 2nd Rev.

Edition, American Mathematical Society, Providence, Rhode Island, 2005.

Narasimhan, R and Nievergelt, Y., Complex Analysis in One Variable, 2nd Edition, Birkh¨auser, Boston,

Basel, Stuttgard, 2000.

Needham, T., Visual Complex Analysis, Rep Edition, Oxford University Press, Oxford, 1999.

Nehari, Z., Conformal Mapping, Dover Publications, New York, 1982.

Paliouras, J D and Meadows, D S., Complex Variables for Scientists and Engineers, Facsimile Edition,

Macmillan Coll Div., New York, 1990.

Pierpont, J., Functions of a Complex Variable (Phoenix Edition), Dover Publications, New York, 2005 Schinzinger, R and Laura, P A A., Conformal Mapping: Methods and Applications, Dover Publications,

New York, 2003.

Silverman, R A., Introductory Complex Analysis, Dover Publications, New York, 1984.

Spiegel, M R., Schaum’s Outline of Complex Variables, McGraw-Hill, New York, 1968.

Sveshnikov, A G and Tikhonov, A N., The Theory of Functions of a Complex Variable, Mir Publishers,

Moscow, 1982.

Wunsch, D A., Complex Variables with Applications, 2nd Edition, Addison Wesley, Boston, 1993.

Integral Transforms

11.1 General Form of Integral Transforms Some

Formulas

11.1.1 Integral Transforms and Inversion Formulas

Normally an integral transform has the form

2

f (λ) =

 b

The function 2f(λ) is called the transform of the function f (x) and ϕ(x, λ) is called the

kernel of the integral transform The function f (x) is called the inverse transform of 2 f (λ) The limits of integration a and b are real numbers (usually, a =0, b = ∞ or a = –∞, b = ∞).

For brevity, we rewrite formula (11.1.1.1) as follows: 2f (u) = L{f (x)}

General properties of integral transforms (linearity):

L{kf (x)}= k L{f(x)},

L{f (x) g(x)}=L{f(x)} L{g(x)}

Here, k is an arbitrary constant; it is assumed that integral transforms of the functions f (x) and g(x) exist.

In Subsections 11.2–11.6, the most popular (Laplace, Mellin, Fourier, etc.) integral transforms are described These subsections also describe the corresponding inversion formulas, which normally have the form

f (x) =



C ψ(x, λ) 2 f (λ) dλ (11.1.1.2)

and make it possible to recover f (x) if 2 f (λ) is given The integration path C can lie either

on the real axis or in the complex plane

In many cases, to evaluate the integrals in the inversion formula (11.1.1.2)—in particular,

to find the inverse Laplace, Mellin, and Fourier transforms — methods of the theory of functions of a complex variable can be applied, including the residue theorem and the Jordan lemma, which are briefly outlined below in Subsection 11.1.2

11.1.2 Residues Jordan Lemma

11.1.2-1 Residues Calculation formulas

The residue of a function f (z) holomorphic in a deleted neighborhood of a point z = a (thus, a is an isolated singularity of f ) of the complex plane z is the number

res

z=a f (z) =

1

2πi



C ε

f (z) dz, i2 = –1,

where C ε is a circle of sufficiently small radius ε described by the equation|z – a|= ε.

435

436 INTEGRALTRANSFORMS

If the point z = a is a pole of order n* of the function f (z), then we have

res

z=a f (z) =

1

(n –1)!z→alim

d n–1

dx n–1



(z – a) n f (z)

For a simple pole, which corresponds to n =1, this implies

res

z=a f (z) = lim z→a



(z – a)f (z)

If f (z) = ϕ(z)

ψ(z) , where ϕ(a) ≠ 0 and ψ(z) has a simple zero at the point z = a, i.e., ψ(a) =0and ψ  z (a)≠ 0, then

res

z=a f (z) =

ϕ(a)

ψ  z (a).

11.1.2-2 Jordan lemma

If a function f (z) is continuous in the domain|z| ≥R0, Im z≥α, where α is a chosen real

number, and if lim

z→∞ f (z) =0, then

lim

R→∞



C R

e iλz f (z) dz =0

for any λ >0, where C Ris the arc of the circle|z|= R that lies in this domain.

 For more details about residues and the Jordan lemma, see Paragraphs 10.1.2-7 and

10.1.2-8

11.2 Laplace Transform

11.2.1 Laplace Transform and the Inverse Laplace Transform

11.2.1-1 Laplace transform

The Laplace transform of an arbitrary (complex-valued) function f (x) of a real variable x (x≥ 0) is defined by

2

f(p) =

 ∞

0 e

where p = s + iσ is a complex variable.

The Laplace transform exists for any continuous or piecewise-continuous function satisfying the condition |f (x)| < M e σ0x with some M >0 and σ0 ≥ 0 In the following,

σ0 often means the greatest lower bound of the possible values of σ0in this estimate; this

value is called the growth exponent of the function f (x).

For any f (x), the transform 2 f (p) is defined in the half-plane Re p > σ0 and is analytic there

For brevity, we shall write formula (11.2.1.1) as follows:

2

f (p) =L5f (x)6

, or f(p) =2 L5f (x), p6.

* In a neighborhood of this point we have f (z)≈const (z – a)–n.