化工應用數學 授課教師： 郭修伯 Lecture 5 Solution by series (skip) Complex algebra

Infinite series They can be accepted as solutions if they are convergent. –As n  , S n  S (some finite number), the series is “convergent”. –As n  , S n  ± , the series is “divergent”. –In other cases, the series is “oscillatory”.

Properties of infinite series If a series contains only positive real numbers or zero, it must be either convergent or divergent. If a series is convergent, then u n  0, as n  . If a series is absolutely convergent, then it is also convergent. –If the series is convergent, it is absolutely convergent.

Power Series A power series about x 0 is: Not every differential equation can be solved using power series method. This method is valid if the coefficient functions in the differential equation are analytical at a point Taylor series: –Maclaurin series (about zero) Frobenius series:

Important topics in “series” Method of Frobenius –The differential equation Bessel’s equation –The equation arises so frequently in practical problems that the series solutions have been standardized and tabulated. can be solved by putting

Special Functions Bessel’s equation of order –occurs in studies of radiation of energy and in other contexts, particularly those in cylindrical coordinates –Solutions of Bessel’s equation when 2 is not an integer when 2 is an integer –when = n + 0.5

Complex algebra r  x y Properties : De Moivre’s theorem: For all rational values of n, Note:  is not included!

Hyperbolic functions Complex numbers and Trigonometric-exponential identities

Derivatives of a complex variable Consider the complex variable to be a continuous function, and let and. Then the partial derivative of w w.r.t. x, is: or Similarily, the partial derivative of w w.r.t. y, is: or Cauchy-Riemann conditions They must be satisfied for the derivative of a complex number to have any meaning.

Analytic functions A function of the complex variable is called an analytic or regular function within a region R, if all points z 0 in the region satisfies the following conditions: –It is single valued in the region R. –It has a unique finite value. –It has a unique finite derivative at z 0 which satisfies the Cauchy-Riemann conditions Only analytic functions can be utilised in pure and applied mathematics.

If w = z 3, show that the function satisfies the Cauchy-Riemann conditions and state the region wherein the function is analytic. Cauchy-Riemann conditions Satisfy! Also, for all finite values of z, w is finite. Hence the function w = z 3 is analytic in any region of finite size. (Note, w is not analytic when z = .)

If w = z -1, show that the function satisfies the Cauchy-Riemann conditions and state the region wherein the function is analytic. Cauchy-Riemann conditions Satisfy! Except from the origin For all finite values of z, except of 0, w is finite. Hence the function w = z -1 is analytic everywhere in the z plane with except of the one point z = 0. ?

At the origin, y = 0 At the origin, x = 0 As x tends to zero through either positive or negative values, it tends to negative infinity. As y tends to zero through either positive or negative values, it tends to positive infinity. Consider half of the Cauchy-Riemann condition, which is not satisfied at the origin. Although the other half of the condition is satisfied, i.e.

Singularities We have seen that the function w = z 3 is analytic everywhere except at z =  whilst the function w = z -1 is analytic everywhere except at z = 0. In fact, NO function except a constant is analytic throughout the complex plane, and every function except of a complex variable has one or more points in the z plane where it ceases to be analytic. These points are called “singularities”.

Types of singularities Three types of singularities exist: –Poles or unessential singularities “single-valued” functions –Essential singularities “single-valued” functions –Branch points “multivalued” functions

Poles or unessential singularities A pole is a point in the complex plane at which the value of a function becomes infinite. For example, w = z -1 is infinite at z = 0, and we say that the function w = z -1 has a pole at the origin. A pole has an “order”: –The pole in w = z -1 is first order. –The pole in w = z -2 is second order.

The order of a pole If w = f(z) becomes infinite at the point z = a, we define: where n is an integer. If it is possible to find a finite value of n which makes g(z) analytic at z = a, then, the pole of f(z) has been “removed” in forming g(z). The order of the pole is defined as the minimum integer value of n for which g(z) is analytic at z = a. 比如： 在原點為 pole, (a=0) 則 n 最小需大於 1 ，使得 w 在原點的 pole 消失。 Order = 1 什麼意思呢？ 在 0 和 a 各有一個 pole ，則 w 在 0 這個 pole 的 order 為 3 在 a 這個 pole 的 order 為 4

Essential singularities Certain functions of complex variables have an infinite number of terms which all approach infinity as the complex variable approaches a specific value. These could be thought of as poles of infinite order, but as the singularity cannot be removed by multiplying the function by a finite factor, they cannot be poles. This type of sigularity is called an essential singularity and is portrayed by functions which can be expanded in a descending power series of the variable. Example: e 1/z has an essential sigularity at z = 0.

Essential singularities can be distinguished from poles by the fact that they cannot be removed by multiplying by a factor of finite value. Example: infinite at the origin We try to remove the singularity of the function at the origin by multiplying z p It consists of a finite number of positive powers of z, followed by an infinite number of negative powers of z. All terms are positive It is impossible to find a finite value of p which will remove the singularity in e 1/z at the origin. The singularity is “essential”.

Branch points The singularities described above arise from the non-analytic behaviour of single-valued functions. However, multi-valued functions frequently arise in the solution of engineering problems. For example: zw For any value of z represented by a point on the circumference of the circle in the z plane, there will be two corresponding values of w represented by points in the w plane.

and Cauchy-Riemann conditions in polar coordinates when 0    2  A given range, where the function is single valued: the “branch” The particular value of z at which the function becomes infinite or zero is called the “branch point”. The origin is the branch point here.

A function is only multi-valued around closed contours which enclose the branch point. It is only necessary to eliminate such contours and the function will become single valued. –The simplest way of doing this is to erect a barrier from the branch point to infinity and not allow any curve to cross the barrier. –The function becomes single valued and analytic for all permitted curves. Branch point

Barrier - branch cut The barrier must start from the branch point but it can go to infinity in any direction in the z plane, and may be either curved or straight. In most normal applications, the barrier is drawn along the negative real axis. –The branch is termed the “principle branch”. –The barrier is termed the “branch cut”. –For the example given in the previous slide, the region, the barrier confines the function to the region in which the argument of z is within the range -  <  < .

The successive values of a complex variable z can be represented by a curve in the complex plane, and the function w = f (z) will have particular value at each point on this curve.

Integration of functions of complex variables The integral of f(z) with respect to z is the sum of the product f M (z)  z along the curve in the complex plane: where f M (z) is the mean value of f(z) in the length  z of the curve; and C specifies the curve in the z plane along which the integration is performed.

When w and z are both real (i.e. v = y = 0): This is the form that we have learnt about integration; actually, this is only a special case of a contour integration along the real axis.

Cauchy’s theorem If any function is analytic within and upon a closed contour, the integral taken around the contour is zero.

If KLMN represents a closed curve and there are no singularities of f(z) within or upon the contour, the value of the integral of f(z) around the contour is: Since the curve is closed, each integral on the right-hand side can be restated as a surface integral using Stokes’ theorem: But for an analytic function, each integral on the right-hand side is zero according to the Cauchy-Riemann conditions Stokes’ theorem

Integral of f(z) between two points The value of an integral of f(z) between two points in the complex plane is independent of the path of integration, provided that the function is analytic everywhere within the region containing all of the paths. P Q

Show that the value of  z 2 dz between z = 0 and z = 8 + 6i is the same whether the integration is carried out along the path AB or around the path ACDB. A C B D The path of AB is given by the equation: Consider the integration along the curve ACDB Along AC, x = 0, z = iy Along CDB, r = 10, z = 10e i  Independent of path

Evaluate around a circle with its centre at the origin Let z = re i  Although the function is not analytic at the origin, Evaluate around a circle with its centre at the origin Let z = re i 

Cauchy’s integral formula  C a A complex function f(z) is analytic upon and within the solid line contour C. Let a be a point within the closed contour such that f(z) is not zero and define a new function g(z): g(z) is analytic within the contour C except at the point a (simple pole). If the pole is isolated by drawing a circle  around a and joining  to C, the integral around this modified contour is 0 (Cauchy’s theorem). The straight dotted lines joining the outside contour C and the inner circle  are drawn very close together and their paths are synonymous.

Since integration along them will be in opposite directions and g(z) is analytic in the region containing them, the net value of the integral along the straight dotted lines will be zero: Let the value of f(z) on  be ; where  is a small quantity. 0, where  is small Cauchy’s integral formula: It permits the evaluation of a function at any point within a closed contour when the value of the function on the contour is known.

The theory of residues The theory of residues is an extension of Cauchy’s theorem for the case when f(z) has a singularity at some point within the contour C.

If a coordinate system with its origin at the singularity of f(z) and no other singularities of f(z). If the singularity at the origin is a pole of order N, then: will be analytic at all points within the contour C. g(z) can then be expanded in a power series in z and f(z) will thus be: Laurent expansion of the complex function The infinite series of positive powers of z is analytic within and upon C and the integral of these terms will be zero by Cauchy’s theorem. the residue of the function at the pole If the pole is not at the origin but at z 0

Evaluate around a circle centred at the origin If |z| < |a|, the function is analytic within the contour Cauchy’s theorem If |z| > |a|, there is a pole of order 3 at z = a within the contour. Therefore transfer the origin to z = a by putting = z - a. non-zero term, residue =

Evaluation of residues without the Laurent Expansion The complex function f(z) can be expressed in terms of a numerator and a denominator if it has any singularities: If a simple pole exits at z = a, then g(z) = (z-a)G(z) Laurent expansion multiply both sides by (z-a) azaz

Evaluate the residues of Two poles at z = 3 and z = - 4 The residue at z = 3: B 1 = 3/(3+4) = 3/7 The residue at z = - 4: B 1 = - 4/( ) = 4/7 Evaluate the residues of Two poles at z = iw and z = - iw The residue at z = iw: B 1 = e iw /2iw The residue at z = - iw: B 1 = -e iw /2iw

If the denominator cannot be factorized, the residue of f(z) at z = a is indeterminate L’Hôpital’s rule Evaluate around a circle with centre at the origin and radius |z| <  /n

Evaluation of residues at multiple poles If f(z) has a pole of order n at z = a and no other singularity, f(z) is: where n is a finite integer, and F(z) is analytic at z = a. F(z) can be expanded by the Taylor series: Dividing throughout by (z-a) n The residue at z = a is the coefficient of (z-a) -1 The residue at a pole of order n situated at z = a is:

Evaluate around a circle of radius |z| > |a|. has a pole of order 3 at z = a, and the residue is: