PHYS 3900 Homework Toolbox V

Transcription

1 PHYS 3900 Homework Toolbox V Complex Numbers, Functions and Algebra Integer: m, n. Real: a, b, r, s, t, x, y, r, θ, φ. Complex: z, w, p, q. (A) Arithmetic and Elementary Algebra. z w = z w, z 2 = z z, z = z, (zw) = z w, z w = ( z w z w, ) z =, w z w = zw w 2, Re(z) = 1 2 (z + z ), Im(z) = 1 2i (z z ), z 2 = [Re(z)] 2 + [Im(z)] 2, Re(az + bw) = a Re(z) + b Re(w), Im(az + bw) = a Im(z) + b Im(w), Re(iz) = Im(z), Im(iz) = + Re(z), z m z n = z m+n z m, z = n zm n, z m w m = (zw) m z m ( z ) m, w =, m w For z = x + iy: e z = e x e iy = e x cos(y) + ie x sin(y), e z = e x, e z / e z = e iy. e iθ = cos(θ) + i sin(θ), e iθ e iφ = e i(θ+φ) e iθ ( ) m, e = iφ ei(θ φ), e iθ = e imθ, e iz = cos(z) + i sin(z), e z e w = e z+w e z ( ) m, e = w ez w, e z = e mz, cos(θ) = Re[e iθ ] = Im[i e iθ ] = 1 2 (eiθ +e iθ ), sin(θ) = Im[e iθ ] = Re[ i e iθ ] = 1 2i (eiθ e iθ ), cos(z) = 1 2 (eiz +e iz ), sin(z) = 1 2i (eiz e iz ), tan(z) = sin(z) cos(z), cos(z) cot(z) = sin(z), ( ) e iθ ( = e iθ, ) e z = e (z ) ( ) ( ), sin(z) = sin(z ), cos(z) = cos(z ), i 2 = 1, i = 1 i, e2miπ = 1, e (2m±1)iπ = e ±iπ = 1, e (2m± 1 2 )iπ = e ±i π 2 = ±i. (B) Conversion between Cartesian and Polar Coordinates for z = x + iy = re iθ. z = x iy = re iθ, x = r cos(θ), y = r sin(θ), r = z = x+iy = x 2 + y 2 0, y x = tan(θ), x y = cot(θ), y x2 + y 2 = sin(θ), x x2 + y 2 = cos(θ), +1 if y > 0, ( x ) θ = sgn(y) arccos + 2mπ, sgn(y) = 1 if y < 0, x2 + y 2 0 if y = 0. 1

2 ( arcsin y ( θ = π arcsin ( π arcsin x 2 +y 2 ) ) y x 2 +y 2 + 2mπ if x 0 and x 2 + y 2 > 0, ) y x 2 +y 2 + 2mπ if x < 0 and y 0, + 2mπ if x < 0 and y < 0. ( ) y arctan + 2mπ if x > 0, x ( ) y π + arctan + 2mπ if x < 0 and y 0, x θ = ( ) y π + arctan + 2mπ if x < 0 and y < 0, x sgn(y) + 2mπ if x = 0 and y 0. π 2 To get θ in the standard interval, meaning: θ ( π, +π] or π < θ +π, set m = 0 in the foregoing three eqs. This particular choice of θ is called the argument of z, denoted by the argument function arg(z) := θ ( π, +π]. If z = 0, then θ is undefined and we set arg(z) := 0. (C) Complex roots, logarithms, powers, exponents and quadratic equations Given complex z = re iθ with polar coordinates r z 0 and angle θ arg(z), defined uniquely by restriction to the standard interval θ ( π, +π]. We then define: The standard complex n-th root of z for integer n > 0: n c z := n r e iθ/n ; Re ( n ) c z = n r cos(θ/n) ; Im ( n ) c z = n r sin(θ/n). So, if w = c n z then w n = z. The definition of the complex n-th root is consistent with the conventional real n-th root: n c z = n z 0 for real z 0 However, the complex n-th root function does not generally obey the rules n c zw = ( n )( c z n ) c w or n c z/w = ( n ) ( c z / n ) c w for arbitrary complex z and w with w 0. The standard complex natural logarithm of z for z 0: ln c (z) := ln( z ) + i arg(z) ; Re [ ln c (z) ] = ln( z ) ; Im [ ln c (z) ] = arg(z). So, if w = ln c (z) then e w = z. The definition of the complex natural logarithm is consistent with the conventional real natural logarithm: ln c (z) = ln(z) for real z > 0. However, the complex natural logarithm function does not generally obey the rules ln c (zw)= ln c (z) + ln c (w) or ln c (z/w) = ln c (z) ln c (w) or ln c (z m ) = m ln c (z) for arbitrary non-zero complex z and w and integer m. 2

3 The standard complex p-th power of a complex base z for arbitrary complex exponent p = s + it with s := Re(p) and t := Im(p): z p c := e p lnc(z) = e [s ln( z ) t θ]+i[s θ+t ln( z )] for z 0 and θ = arg(z). Re ( z p c ) = e [s ln( z ) t θ] cos[s θ + t ln( z )], Im ( z p c ) = e [s ln( z ) t θ] sin[s θ + t ln( z )] For z = 0, we set 0 p c := 0 if p > 0 and 0 0 c := 1. The definition of the complex p-th power is consistent with the conventional p-th power for real base z > 0 and real exponents p: z p c = z p for real z > 0 and real p. The definition of the complex p-th power is also consistent with the conventional p-th power for integer exponents p = m: m n=1 z = z z... z (m times), for integer m > 0 and complex z 0 z m c = z m = 1, for integer m = 0 and complex z 0 m 1 n=1 = ( m times), for integer m < 0 and complex z 0 z z z z The definition of the complex p-th power is also consistent with the conventional relation between fractional powers and roots for the case of rational exponents, p = m/n, with integer m and n and n > 0: z (m/n) c = [ c n (z) ] m, for integers m and n with n > 0. The complex power obeys the exponent addition and integer multiplication rules: z (p+q) c = ( z p c ) ( z q c ) ; z (mp) c = ( z p c ) m, for complex z, p and q, z 0, integer m. However, the complex power function with complex base and complex exponents does not generally obey the rule z (qp) c = ( z p c ) q for arbitrary non-integer values of q. Also, it does not generally obey the rule (zw) p c = ( z p c )( w p c ) for arbitrary complex p and non-zero complex z and w. The complex quadratic equation Az 2 + Bz + C = 0 with complex coefficients A 0, B and C always has two complex roots, z + and z, written in terms of the above-defined complex square root function, c..., as: z ± = B ± c B2 4AC 2A The corresponding quadratic polynomial can thus always be factorized by Az 2 + Bz + C = A (z z + ) (z z ) for all complex z The two roots become degenerate, z + = z, if and only if B 2 = 4AC; else z + z. 3

4 (D) Complex polynomial factorization and reduction. Given is a complex polynomial P (z) of complex variable z, of degree (a.k.a. order) N, written in standard form with arbitrary complex coefficients a 0, a 1,..., a N and with a N 0: P (z) = a 0 + a 1 z + a 2 z a N z N Fundamental Theorem of Algebra. There exists an N-tuple of complex numbers, denoted by (z 1, z 2,...z N ), so that P (z) can be written in the following factorized form: P (z) = a N (z z 1 ) (z z 2 )... (z z N ) for all complex z. Obviously, each z k for k = 1, 2,..., N is a root of P (z), meaning: P (z 1 ) = 0, P (z 2 ) = 0,..., P (z N ) = 0. Also, obviously z 1, z 2,...z N are the only possible roots of P (z), i.e., if P (z c ) = 0 for some complex z c, then z c must be one of the z k -values: z c = z k for some k {1, 2,..., N}. The term N-tuple here means: an ordered list of N complex numbers, labeled by an integer index k, running through the values k = 1, 2,..., N. This implies, and allows for the possibility, that some of the z k -entries in this root list may have the same value. Such roots occuring more than once in the root list are sometimes referred to as degenerate roots; and the number of times a root value occurs in the list is referred to as the degeneracy or the multiplicity of that root. So, for example, for degree N = 6, some polynomial P (z) may have a root list (z 1, z 2, z 3, z 4, z 5, z 6 ) = ( 5, 2i, 1 + i, 2i, 5, 2i). So the root 5 has multiplicity 2 here, since 5 occurs 2 times in the list: z 1 = z 5 = 5. Likewise, the root 2i has multiplicity 3, since 2i occurs 3 times in the list: z 2 = z 4 = z 6 = 2i. Lastly, the root 1+i has multiplicity 1, since 1+i occurs only 1 time in the list: z 3 = 1+i. So, in factorized form, this polynomial P (z) can be written as P (z) = a 6 (z + 5)(z + 2i)(z 1 i)(z + 2i)(z + 5)(z + 2i). Using multiplicities, P (z) can be written more compactly as P (z) = a 6 (z + 5) 2 (z + 2i) 3 (z 1 i). Remainder Theorem. For any complex constant z c, P (z) can be divided by a linear divisor, (z z c ), resulting in a quotient polynomial Q(z) of degree (N 1) and a remainder term of the form P (z c )/(z z c ), as follows: P (z) = Q(z) + P (z c) z z c z z c for all complex z z c or equivalently P (z) = (z z c ) Q(z) + P (z c ) for all complex z. The quotient polynomial Q(z) can be written in the standard form Q(z) = b 0 + b 1 z + b 2 z b N 1 z N 1 4

5 with coefficents b 0, b 1,..., b N 1 that are obtained from z c and a 0, a 1,...a N by: or in general form: b N 1 = a N b N 2 = a N 1 + a N z c b N 3 = a N 2 + a N 1 z c + a N z 2 c... b 1 = a 2 + a 3 z c a N 1 z N 3 c b 0 = a 1 + a 2 z c + a 3 z 2 c... + a N 1 z N 2 c b j = a j+1 + a j+2 z c a N 1 z N j c Factor Theorem. If z c is a root of P (z), i.e., if then P (z) can be factorized in the form + a N z N 2 c + a N z N 1 c + a N z N j 1 c for j = 0, 1, 2,..., (N 1). P (z c ) = 0, P (z) = (z z c ) Q(z) for all acomplex z. Here Q(z) is a polynomial of degree (N 1) and its coefficients b 0, b 1,...b N 1 are obtained from z c and the P (z)-coefficients, a 0,..., a N, by the Remainder Theorem formulae above. Also, by the Fundamental Theorem, if z c is a root of P (z) it must coincide with one of the entries z k in the root list of P (z), (z 1,..., z N ). Also, for simplicity we can assume that the arbitrarily choosen ordering of the list is such that k = 1, meaning: z c = z 1. Then Q(z) = P (z)/(z z c ) = a N (z z 2 )(z z 3 )... (z z N ). In other words, the root list of Q(z) is (z 2, z 3,..., z N ): it has the same entries as the root list of P (z), except that the entry z 1 has been removed. Therefore, all the other roots of P (z), excluding z c z 1, can be found by solving for the roots of Q(z). This is one of the main applications of the foregoing Theorems: they allow us to search for remaining unknown roots of P (z) by solving for the roots of Q(z) which is a polynomial of a lower degree than P (z) itself. So, for example, if P (z) is a cubic polynomial, with a root list (z 1, z 2, z 3 ), and if we already know one of its roots, z c = z 1, then we can reduce P (z) to Q(z). That is, the above Remainder Theorem formulae allow us to calculate the Q(z)-coefficients b 0, b 1, b 2 from the P (z)-coefficients a 1, a 2, a 3 and from the known z c = z 1. Q(z) is now a quadratic and has the root list (z 2, z 3 ). We can thus find the remaining two roots of P (z), z 2 and z 3, by finding the two roots of Q(z), i.e., by solving the quadratic equation Q(z) = 0. (E) Derivative and anti-derivative of f(t) := e wt for complex w, real t. d dt f(t) d ) (e wt = we wt, f(t)dt e wt dt = 1 dt w ewt for w 0. 5

6 Complex Representation of Oscillations (A) Complex Amplitudes Alternating ( sinusoidally oscillating) currents Ĩ(t) and voltages Ṽ (t) in electrical circuits, as well as any other sinusoidally oscillating quantities in physics, engineering, etc., can be represented and analyzed in terms of complex amplitudes. Assume Ĩ(t) and Ṽ (t) both oscillate with angular frequency ω, corresponding oscillation frequency f = ω/(2π), oscillation period T = 2π/ω; with respective phase shifts φ I and φ V ; and with respective positive, real amplitudes, I o > 0 and V o > 0: Ĩ(t) = I o cos(ωt φ I ), Ṽ (t) = V o cos(ωt φ V ). Then Ĩ(t) and Ṽ (t) can be written as: Ĩ(t) = Re ( Ie iωt), Ṽ (t) = Re ( V e iωt), with complex amplitudes, I and V, defined by I := I o e iφ I = I o cos(φ I ) ii o sin(φ I ), V := V o e iφ V = V o cos(φ V ) iv o sin(φ V ). In the foregoing representation of Ĩ(t), the time-dependent complex current, Ieiωt, can be visualized as a vector in the complex plane, having a length I o and an angle ωt φ I to the real axis. As a function of time t, this vector is rotating around in the complex plane, at an angular velocity ω, in counter-clockwise direction, with the tip of the vector tracing out a circle of radius I o. The time-dependent real alternating current, Ĩ(t), is simply the projection of the rotating complex-plane vector, Ie iωt, onto the real axis. Likewise, the time-dependent real alternating voltage, Ṽ (t), is the projection of the rotating complex-plane vector, V eiωt, onto the real axis, with the tip of the rotating vector tracing out a circle of radius V o. Real and complex amplitudes are related by: I o = I = (ReI) 2 + (ImI) 2, V o = V = (ReV ) 2 + (ImV ) 2. The (real) phase angles and complex amplitudes are related by: tan(φ I ) = ImI ReI, cos(φ I) = ReI I, sin(φ I ) = ImI I, tan(φ V ) = ImV ReV, cos(φ V ) = ReV V, sin(φ V ) = ImV V. (B) Identity and Superposition of Oscillations Identity of Oscillations. Consider two sinusoidally osillating quantities, of the same angular frequency ω 0, given by Ṽ (t) and Ũ(t), Ṽ (t) = V o cos(ωt φ V ) = Re ( V e iωt) and Ũ(t) = U o cos(ωt φ U ) = Re ( Ue iωt), 6

7 with real, positive amplitudes V o and U o ; phase shifts, φ V and φ U, each within the standard interval, meaning: φ V, φ U [ π, π); and complex amplitudes V :=V o e iφ V and U :=U o e iφ U. Then the following three statements are equivalent: (1) Ṽ (t) = Ũ(t) for all times t. (2) V = U. (3) V o = U o and φ V = φ U. Statement (2) follows from (1), like this: From Ṽ (t)=re( V e iωt), get dṽ (t)/dt=re( iωv e iωt). Thus, at time t = 0, where e iωt = 1, we have Ṽ (0) = Re( V ) and dṽ (0)/dt = Re( iωv ) = ωim ( V ). So, Re ( V ) =Ṽ (0) and Im( V ) = (1/ω) dṽ (0)/dt. Likewise, Re( ) U =Ũ(0) and Im ( U ) = (1/ω) dũ(0)/dt. But from (1) also follows dṽ (t)/dt = dũ(t)/dt and, at time t = 0, Ṽ (0) = Ũ(0), and dṽ (0)/dt = dũ(0)/dt. Hence, Re( V ) = Re ( U ) and Im ( V ) = Im ( U ) ; and hence V = U. Statement (3) follows from (2), since V o = V and φ V = arg(v ); and likewise U o = U and φ U = arg(u). Thus, from statement (2), we conclude V o =U o and φ V = φ U. Statement (1) follows from (3), since Ṽ (t) = V o cos(ωt φ V ) and Ũ(t) = U o cos(ωt φ U ). Thus, from statement (3), we conclude Ṽ (t)=ũ(t) at all times t. Superposition of Oscillations. Consider three sinusoidally osillating quantities, Ṽ (t), Ũ(t) and W (t), of the same angular frequency ω 0, given by Ṽ (t) = V o cos(ωt φ V ) = Re ( V e iωt) Ũ(t) = U o cos(ωt φ U ) = Re ( Ue iωt), W (t) = W o cos(ωt φ W ) = Re ( W e iωt), with real, positive amplitudes V o, U o and W o ; phase shifts, φ V, φ U and φ W, each within the standard interval, meaning: φ V, φ U, φ W [ π, π); and complex amplitudes V := V o e iφ V, U :=U o e iφ U and W :=W o e iφ W. Then the following three statements are equivalent: (1) Ṽ (t) = Ũ(t) + W (t) for all times t. (2) V = U + W. (3) V o = U + W and φ V = arg(u + W ). (C) Complex Impedance In linear circuit elements current and voltage are proportional to each other. The complex amplitudes, I and V, of an alternating current through the element, Ĩ(t) = Re(Ieiωt ), and of the alternating voltage drop across the element Ṽ (t) = Re(V eiωt ), are then related by a generalized Ohm s Law, such that the so-called complex impedance Z = V I 7

8 is independent of I or V. Z depends only on the nature/design of the circuit element and, in general, on the angular frequency ω. Examples: Complex Impedance Z R of an Ohmic Resistance R. Re[V e iωt ] = Ṽ (t) = RĨ(t) = Re[RIe iωt ], hence V =RI, hence Z R := V I = R. Complex Impedance Z L of an Inductance L. Re[V e iωt ]=Ṽ (t)=l d dtĩ(t)=re[l d dt (Ieiωt )]= Re[iωLIe iωt ], hence V =iωli, hence Z L := V I = iωl. Complex Impedance Z C of a Capacitance C. Alternating charge stored on upper capacitor plate q(t)=cṽ (t)=re(cv eiωt ), hence Re[Ie iωt ]=Ĩ(t)= d q(t)=re[ d (CV dt dt eiωt )]= Re[iωCV e iωt ], hence I =iωcv, hence (D) Impedance Combination Rules Z C := V I = 1 iωc. If two elements X and Y with complex impedances Z X and Z Y in an AC circuit are... Connected in Series, then they can be replaced by a single equivalent impedance Z = Z X + Z Y with I X = I Y = I and V X + V Y = V ; Connected in Parallel, then they can be replaced by a single equivalent impedance Z = ( 1 Z X + 1 Z Y ) 1 with I X + I Y = I and V X = V Y = V. Here, I X and I Y denote the complex amplitudes of the alternating currents in element X and element Y, respectively; and V X and V Y denote the complex amplitudes of the respective alternating voltage drop across the element. I and V denote the complex amplitudes of the total alternating current and total alternating voltage drop, respectively, for the entire combination of both elements (serial or parallel). Generalized Ohm s Law. The complex current and complex voltage drop amplitudes, for either parallel or serial combination of elements, are then related by their respective generalized Ohm s Laws I = V Z, I X = V X Z X, I Y = V Y Z Y or V = ZI, V X = Z X I X, V Y = Z Y I Y. 8

9 Limits and Convergence Criteria (A) Ratio Test (d Alembert Convergence Criterion). Assume T j is a sequence of complex numbers with T j 0, defined for all integers j j o with some starting value j o ; S = is its infinite complex series; and the limit j=j o T j ρ := lim j T j+1 /T j exists or ρ =. Then S is divergent for ρ > 1, including ρ = ; and S is absolutely convergent for ρ < 1. Assume T j is of the form T j (z) = A j (z z o ) m j, defined for some non-zero complex coefficient sequence A j 0, integer exponent sequence m j, and complex variables z and z o. Then S = S(z) is a power series and the condition ρ(z) = 1 defines the boundary of the circular convergence disk of this power series. (B) Squeeze Theorem. If two sequences of real numbers, P j and Q j, obey: 0 P j Q j for all integers j j o with some starting value j o, and if then also: lim Q j = 0, j lim P j = 0. j (C) Exponential-Kills-Power Theorem. Any exponential function e Kx with K > 0 dies out faster (i.e., approaches zero faster) than any power of real x can rise to infinity: lim x + xp e Kx lim x + Likewise, replacing x by integers j > 0, e Kx = 0 for any real p and real K with K > 0. x p lim j + jp e Kj e Kj lim = 0 for any real p and real K with K > 0. j + j p 9

10 (D) Applications to Proofs of Convergence. Using (C), one can show that lim j jp w j = 0 for all complex w with w < 1 and all real p. This is so because one can set K := ln( w ), then write w j = e Kj and note that K > 0 for 0 < w < 1. Hence, by (C), j p w j = j p e Kj 0 for j. Combining this with the Squeeze Theorem (B), one can then show that lim j jp w j j! = 0 for all complex w with w < 1 and all real p > 0. This is so because 0 w j j! w j, hence 0 j p w j j! j p w j when w < 1. Thus, by (B), P j := j p w j j! must converge to 0, as Q j := j p w j converges to 0, for j. (E) Applications to Proofs of Divergence. Using (D), one can show that lim j jp w j = and lim j p w j j! = for all complex w with w > 1 and all real p. j This is so because one can define w := 1/w and p := p, so that 1/[j p w j ] = j p w j, likewise 1/[j p w j j! ] = j p w j j! and also 0 < w = 1/ w < 1 when w > 1. Using (D), one thus can show that the sequences of reciprocals, 1/[j p w j ] and 1/[j p w j j! ], converge to 0 for j when w > 1. Hence, the corresponding sequences themselves, j p w j and j p w j j!, must diverge to + when j, for all complex w with w > 1 and all real p. 10

11 Complex-Variable Limits and Differential Calculus For all the following, assume that U is an open set in the complex plane and z o is a fixed complex number in U. The set of all complex numbers in U excluding z o is denoted by U\{z o } and is then also an open set. Assume further that complex-valued functions f(z) and g(z) are defined for all complex arguments z in U\{z o }. That is, f(z) and g(z) are defined for all z in U, except possibly for z = z o. Function values f(z) and/or g(z) may also be defined for z = z o, but that is not required, unless stated otherwise. (A) Taking Limits in the Complex Plane. Assume that d is a non-zero complex number, pointing at some angle, arg(d) from the postive real axis: d defines a direction in the complex plane. For real values τ and fixed d and z o, let z(τ) := z o + τd. All points z(τ) form a straight line which is parallel to the d-vector and passes through z o when τ = 0. Since U is an open set containing z o, we can find a real positive number, ɛ > 0, so that all z with z z o < ɛ are also contained in U. If we choose τ < ɛ/ d, then z(τ) z o = τd = τ d < ɛ. So, all points z(τ) on the straight line with 0 < τ < ɛ/ d fall inside U; and for all those z(τ), a function value f(z(τ)) is defined for sufficiently small τ arbitrarily close to τ = 0. We can therefore study the limit of f(z(τ)) for τ 0 or τ 0 +, as follows: Suppose we can find a complex value f o such that f(z(τ)) approaches f o, in the sense that the the (real-valued!) distance between f(z(τ)) and f o, i.e., f(z(τ)) f o, approaches zero for τ 0 + : lim τ 0 + f(z(τ)) f o = 0. If such an f o exists, it is unique; we call it the directional limit of f(z) for z approaching z o from direction d and we define: lim f(z(τ)) := f o or lim f(z o + τd) := f o. τ 0 + τ 0 + On the other hand, if such an f o does not exist, then we say that f(z) does not have a directional limit for z approaching z o from direction d. In this manner, the existence and value of a complex-valued directional limit is defined in terms of the process of taking real-valued limits (which we already know how to do!). In general, the directional limit f o, if existent, will depend on the direction d: for different directions d, f o may have different values, now denoted by f o (d). As an example, consider the function f(z) := arg(z) + i z, defined for all complex z, and the directional limits for z approaching z o = 0 along direction d: f o = lim τ 0 + f(z o + τd) = lim τ 0 +[arg(τd) + d τ ] = arg(d). This is so because, in the complex plane, the vector τd points in the same direction as d when τ > 0, hence arg(τd) = arg(d). So, for example, f o (d) = 0 for d = 2, f o (d) = π/4 = 45 o for d = 1 + i, and f o (d) = 3π/4 = 135 o for d = 1 i. Assume the directional limit of f(z) for z approaching z o does exist from all directions, d 0, and its value, f o = lim τ 0 + f(z o + τd), is independent of d. Then we say, for short, that 11

12 f(z) has a complex limit f o for z z o and we define lim f(z) := f o, (independent of d 0). z z o The foregoing limit notation now implies that f o is independent of the direction d from which z o is being approached by z. If on the other hand, the directional limit does not exist for some d; or if it does exist for all d, but does depend on d, then we say that f(z) has no limit for z z o ; or we say that a limit of f(z) for z z o does not exist. (B) Complex-Variable Limit Theorems Limit Algebra. Assuming lim z zo f(z) and lim z zo g(z) exist, all other limits stated below also exist and the following equations hold, with a and b denoting arbitrary complex numbers and n denoting a positive integer: lim Re ( f(z) ) = Re ( lim f(z) ), z z o z zo lim z z o ( af(z)+bg(z) ) = a lim z zo f(z)+b lim z zo g(z), lim e f(z) = e limz zo f(z), z z o lim Im ( f(z) ) = Im ( lim f(z) ), z zo z zo lim cos ( f(z) ) = cos ( lim f(z) ), z zo z zo If lim z zo g(z) 0 and g(z) 0 for all z in U\{z o }, then f(z) lim z z o g(z) = lim z z o f(z) lim z zo g(z), lim z z o ( f(z) ) n = ( lim z zo f(z) ) n, If arg ( lim z zo f(z) ) π and lim z zo f(z) 0 and n 2, then f(z) lim = lim z zo z zo f(z) ( ) ( lim f(z) g(z) = lim f(z) ) ( lim g(z) ), z zo z zo z zo, lim sin ( f(z) ) = sin ( lim f(z) ). z zo z zo ( ) n ( lim g(z) = lim g(z) ) n. z zo z zo lim arg ( f(z) ) = arg ( lim f(z) ), z z o z zo lim z zo ln c ( f(z) ) = lnc ( lim z zo f(z) ), lim z z o n c f(z) = n c lim z zo f(z), ( ) a c lim f(z) = ( lim f(z) ) a c. z zo z zo Sequence-of-Function-Values Theorem. Let (z 1, z 2, z 3,...) be an infinite sequence of complex numbers, in U\{z o }, which converges to, but never attains, the value z o U, i.e., lim z n = z o and z n z o for all n = 1, 2, 3,... n Assume now that lim z zo f(z) exists. Then the sequence of function values, f(z n ), exists, converges and its limit is the same for all such sequences: lim f(z n) = n lim f(z). z zo Assume, vice versa, that the sequence of function values, f(z n ), convergenes to the same limit, lim n f(z n ), for all convergent sequences (z 1, z 2, z 3,...) in U\{z o } with lim n z n = z o. Then the limit of f(z) for z z o exists and its value is lim f(z) = z z o 12 lim f(z n ). n

13 (C) Continuity, Complex Differentiability, Differentiation Rules Assume in the following that f(z) and g(z) are defined for all z in the open set U, excluding none. Continuity. For z o in U, the function f(z) is said to be continuous at z o if its limit for z z o exists and is the same as the function evaluated at z o : lim f(z) = f(z o ). z z o The function f(z) is said to be continuous on U if it is continuous at all z in U. Complex differentiability, holomorphicity. For z o in U, f(z) is said to be complex differentiable (CD) at z o, if the limit f (z o ) := lim z zo f(z) f(z o ) z z o f(z o + z) f(z o ) = lim z 0 z exists. Analous to derivatives of functions of real variables, we then call f (z o ) the (complex) derivative or differential quotient of f(z) at z o. Analous to real variable functions, we then also use the differential quotient notation d dz f(z o) f (z o ). If f(z) is complex differentiable at all z in U, we say that f(z) is complex differentiable (CD) on U or, synonymously, we say that f(z) is holomorphic on U. Complex differentiability (CD) implies continuity. If f(z) is CD at z o, then f(z) is also continuous at z o. If f(z) is CD (holomorphic) on U, then f(z) is also continuous on U. But the converse is not necessarily true: the function f(z) := Re(z) is continuous everywhere in the complex plane; and it is complex differentiable nowhere in the complex plane. (Check the Cauchy-Riemann conditions, given below, to verify the latter!) Higher order complex derivatives. Higher order complex-variable derivatives of f(z) are defined for all z in U analgous to higher-order derivatives in real-variable calculus: f (0) (z) := f(z), f (1) (z) d dz f(z) := d dz f (0) f (0) (z + z) f (0) (z) (z) = lim z 0 z f (2) (z) d2 dz f(z) := d 2 dz f (1) f (1) (z + z) f (1) (z) (z) = lim z 0 z f (3) (z) d3 dz f(z) := d 3 dz f (2) (z) = lim...and so on z 0 f (2) (z + z) f (2) (z) z,,, 13

14 Or, written recursively for n = 1, 2, 3,...: f (0) (z) := f(z), f (n) (z) dn dz f(z) := d n dz f (n 1) f (n 1) (z + z) f (n 1) (z) (z) = lim z 0 z The existence of higher-order derivatives f (n) (z) thus requires that all the foregoing complexvariable limits ( z 0) do exist for all z in U, up to the order n considered. As discussed in more detail below, the existence of f (1) (z) f (z) for all z in U does in fact guarantee the existence of all higher-order derivatives f (n) (z) for all z in U. So, if f(z) is holomorphic on U, i.e., has a 1 st derivative, f (z), for all z in U, then f(z) also has an n th derivative, f (n) (z), for any order n 2 and for all z in U. Complex differentiation rules. If f(z) and g(z) are CD at some z, then so are the superposition function af(z) + bg(z) with arbitrary complex constants a and b; the product function f(z)g(z); and, provided g(z) 0 for all z in U, the quotient function f(z)/g(z), with the following differentiation rules: ( ) (z) af + bg = af (z) + bg (z) (Superposition Rule), ( fg ) (z) = f (z)g(z) + f(z)g (z) (Product Rule), ( f g ) (z) = f (z)g(z) f(z)g (z) ( g(z) ) 2 (Quotient Rule). Let a function h(w) be defined for all w in an open set V where V contains the range of the function g(z) defined on the open set U, meaning: for every z in U, the function value w g(z) is contained in V. Then we can define the compound function f(z) := h ( g(z) ), for all z in U, and f(z) is CD at z if g(z) is CD at z and if h(w) is CD at w = g(z), with f (z) = h ( g(z) ) g (z) (Chain Rule). Derivatives of elementary functions in the complex plane. For arbitrary complex a, integer n, and all complex z with exceptions noted: d dz (zn ) = nz n 1 (z 0 if n < 0), d dz (ez ) = e z, d d (sin z) = cos z, dz (cos z) = sin z dz Hence, using quotient rule and tan z = sin z/ cos z and cot z = cos z/ sin z, we have: d dz (tan z) = 1 cos 2 z d 1 (cot z) = dz sin 2 z (z mπ + π/2 with integer m), (z mπ with integer m). 14

15 Also, using chain rule and a z c = e z lnc a, we have for a 0: d ) (a z c dz = ln c (a) a z c. Also, if arg(z) π and z 0 (i.e., if z is neither zero nor negative real), then d ( lnc z ) = 1 dz z, d ) (z a c = a c dz z za = az a 1 c d ( ), n c z = 1 n z. dz nz c (D) Cauchy-Riemann Differentiability (Holomorphicity) Conditions Let the function f(z) be defined for all z in the open set U and let u(x, y) := Re ( f(x + iy) ), v(x, y) := Im ( f(x + iy) ) for all real x and y with z = x + iy in U (i.e., x = Re(z) and y = Im(z) here). Then f(z) is holomorphic (i.e. CD) on U if and only if u(x, y) and v(x, y) have partial derivatives with respect to both real variables x and y, obeying the following so-called Cauchy- Riemann differential equations for all x, y with x + iy in U: x u(x, y) = + y v(x, y) and u(x, y) = v(x, y). y x If f(z) is holomorphic on U, its complex derivative f (z) at z = x + iy can be expressed in terms of the partial derivatives of u(x, y) and v(x, y) by any of these four equivalent equations: df dz = u x + i v x = v y i u y = u x i u y = v y + i v x. Taylor and Laurent Series, Singularities, Residues (A) Taylor Series Expansion Circular disks in the complex plane. A circular disk of radius r 0, centered at some complex z o, denoted by U r (z o ), is defined as the set which includes all complex z whose distance from z o falls short of r, as well as z = z o : U r (z o ) = { z C z zo < r or z = z o } We include in this definition the special degenerate cases of (1) r = : U (z o ) = C, i.e., U (z o ) consists of the entire complex plane. (2) r = 0: U 0 (z o ) = {z o }, i.e., U 0 (z o ) consists only of the point z o itself. We also define the circumference of a circular disk centered at z o, denoted by C r (z o ), as the set of all complex z whose distance from z o equals r: C r (z o ) = { z C z zo = r }. 15

16 Note that U r (z o ) is an open set for r > 0, including r = ; but it is not an open set for r = 0. For r > 0, the disk U r (z o ) does not include any points z from its circumference C r (z o ). For r = 0, the disk U 0 (z o ) comprises only z o and coincides with its circumference C 0 (z o ). Convergence disk theorem for power series. Consider a power series around some complex z o n fn(z) := a k (z z o ) k k=0 defined for all integer n 0 with an infinite sequence of complex coefficients a o, a 1, a 2,... The convergence behavior of this series in the limit n defines a so-called radius of convergence R and a corresponding circular convergence disk U R (z o ) by way of the following theorem: There exists a single non-negative real number, R 0, including possibly R =, such that (1) f n (z) is guanranteed to converge as n for all complex z within the circular disk U R (z o ); and (2) f n (z) is guanranteed to diverge as n for all complex z which are neither inside U R (z o ) nor on the circumference C R (z o ). For the special case R =, the series converges for any z in the complex plane. For the special case R = 0, the series converges only for z = z o. If 0 < R < the theorem gives no information about convergence or divergence for any z on the circumference. If z C R (z o ), the series could be either convergent or divergent, depending on z z o and on the sequence of coefficients a 0, a 1, a 2,... Differentiability of power series. Let the function f(z) be defined by a power series around some center z o : f(z) = a k (z z o ) k, k=0 with coefficient sequence, a 0, a 1, a 2,..., and with a non-zero radius of convergence R > 0, including possibly R =. Then f(z) is complex differentiable (CD) for all z in the open circular convergence disk, U R (z o ). Furthermore, for all z in U R (z o ), the derivative function, f (z), can then be obtained by differenting the power series term-by-term: f (z) d dz f(z) = k=0 a k d dz (z z o) k = ka k (z z o ) k 1 = k=1 (j + 1)a j+1 (z z o ) j The foregoing power series expansion of f (z) is then guaranteed to have the same radius of convergence, R, around z o as the power series for f(z). By applying the foregoing differentiability and term-by-term expansion theorem to f (z) and then to f (z), and so on, we thus conclude that all higher-order derivatives of f(z) exist for all z in the open convergence disk U R (z o ). Thus, f(z) is holomorphic on U R (z o ), all its higher-order derivatives f (n) (z) exist, and they are holomorphic on U R (z o ) as well. 16 j=0

17 Unlimited differentiability theorem for holomorphic functions. Let f(z) be CD (holomorphic) on U, i.e, f(z) is CD for all z in U, excluding none. Then its derivative function f (z) is also CD for all z in U. In other words, if f(z) is CD (holomorphic) on U, then f (z) is CD (holomorphic) on U as well. By the foregoing theorem, f (z) must have a derivative, f (z), for all z in U, and f (z), must also be CD (holomorphic) on U, and so on. So, by appyling the foregoing theorem recursively, one can prove that any CD (holomorphic) f(z) on U, has higher-order complex derivative functions f (n) (z) to all orders n: the higher derivatives of f(z), f (0) (z) := f(z), f (1) (z) := d dz f (0) (z), f (2) (z) := d dz f (1) (z),..., f (n) (z) := d dz f (n 1) (z), exist for all orders n. In other words, if f(z) is CD (holomorphic) on U, all the foregoing derivatives f (n) (z) exist for all z in U and all integer n 0. Taylor series expansion theorem for holomorphic functions. A CD (holomorphic) function f(z) on U has a unique power series expansion with non-zero radius of convergence around any center of expansion z o in U. This means that, for any z o in U, one can find a unique sequence of complex coefficents, a 0, a 1, a 2,..., so that f(z) can be written as f(z) = a k (z z o ) k k=0 and this series is guaranteed to have a non-zero radius of convergence R > 0, including possibly R =, i.e.,the foregoing power series converges for all z with z z o < R. Furthermore, the coefficients of this power series are given by a k = 1 k! f (k) (z o ) where f (k) (z o ) is the k-th derivative of f(z), evaluated at the center of expansion z o. The foregoing power series is also referred to as the Taylor series of f(z) around z o. A function f(z) on U which can be expanded into a power series of finite radius of convergence R > 0 around any z o in U is also called an analytic function on U. The Taylor series thereom thus implies that every holomorphic function on U is also an analytic function on U. And vice versa, every analytic function on U is complex differentiable for every z in U. Hence, every analytic function on U is also holomorphic on U. For this reason, the terms holomorphic function, or complex differentiable function, and the term analytic function are often used synonymously in complex-variable calculus. Note that the sequence of Taylor series expansion coefficients, a 0, a 1, a 2,..., as well as the radius of convergence, R, do depend on the center of expansion z o. Also, for given z o, the open circular convergence disk, U R (z o ), may be either fully contained within U; or it may be only partially contained within U; or it may in fact fully contain U. Since U is an open set, we are guaranteed that there exists at least some small finite circular disk U r (z o ), of positive radius r around z o, which is fully contained within both U and U R (z o ): U r (z o ) U U R (z o ) for some real r > 0. 17

18 (B) Laurent Series Expansion Circular rings in the complex plane. An open circular ring of inner radius q 0 and outer radius r > 0 centered around z o, denoted by U q,r (z o ), is defined as the set of all complex numbers z whose distance from z o exceeds q and falls short of r: U q,r (z o ) := { z C q < z zo < r }. We include in this definition the special degenerate cases of (1) q = 0 and 0 < r < : U 0,r (z o ) = U r (z o )\{z o } is an open circular disk U r (z o ) excluding its center point z o. (2) q = 0 and r = : U 0, (z o ) = C\{z o } is the entire complex plane excluding the point z o. (3) q > 0 and r = : U q, (z o ) = C\[U q (z o ) C q (z o )] is the entire complex plane excluding the circular disk of radius q centered around z o and excluding the circumference of that disk. (4) q r: U q,r (z o ) = {} is the empty set: it contains no complex numbers z. Convergence ring theorem for Laurent series. around some complex z o Consider a so-called Laurent series f n (z) := n n a k (z z o ) k + b k (z z o ) k k=0 k=1 defined for all integer n 0 with an infinite sequence of complex a-coefficients, a o, a 1, a 2,... and an infinite sequence of complex b-coefficients, b 1, b 2,... The convergence behavior of this series in the limit n defines a so-called outer radius of convergence R an inner radius of convergence Q and a corresponding circular convergence ring U Q,R (z o ) by way of the following theorem: There exists a pair of non-negative real numbers Q and R, including possibly R =, such that (1) f n (z) is guanranteed to converge as n for all complex z within the circular ring U Q,R (z o ); and (2) f n (z) is guanranteed to diverge as n for all complex z which are neither inside U Q,R (z o ), nor on the inner circumference C Q (z o ), nor on the outer circumference C R (z o ). Of particular interest for applications of the residue theorem, to functions with isolated singularities, are the special cases where 0 = Q < R. For the special case 0 = Q < R <, the series is guaranteed to converge inside the circular disk of radius R around z o, U R (z o ), excluding its center z o. For the special case 0 = Q < R =, the series is guaranteed to converge for all z in the complex plane, excluding the point z o. In the special case 0 Q = R <, the two circumferences C Q (z o ) and C R (z o ) coincide; the series is guaranteed to diverges for all z that are not on that circumference; and the theorem gives no information about convergence or divergence for any z on the circumference. 18

19 If 0 Q < R < the theorem gives no information about convergence or divergence for any z on either the inner or the outer circumference, C Q (z o ) or C R (z o ), respectively. If z C Q (z o ) C R (z o ), the series could be either convergent or divergent, depending on z z o and on the sequences of coefficients a 0, a 1, a 2,... and b 1, b 2,... Likewise, if 0 Q < R = there is no outer circumference and the theorem gives no information about convergence or divergence for any z on the inner circumference, C Q (z o ). If z C Q (z o ), the series could be either convergent or divergent, depending on z z o and on the sequences of coefficients a 0, a 1, a 2,... and b 1, b 2,... The inner and outer convergence radii, Q and R, can be easily related to the convergence radii of the two power series contributing to f n (z), the regular part and the singular part of f n (z), defined, respectively, by so that f [rg] n (u) := n a k u k, k=0 f [sg] n (w) := n b k w k f n (z) = f n [rg] (u) + f n [sg] (w) with u = (z z o ) and w = 1/(z z o ). Let R [rg] and R [sg] denote the respective radii of convergence of f [rg] n k=1 (u) and f n [sg] (w) for n. The outer radius of the Laurent convergence disk, R, is then simply the radius of convergence of f [rg] n : R = R [rg]. The inner radius is basically the reciprocal of the radius of convergence of f [sg] n : Q = 1/R [sg] with the understanding that Q = 0 if R [sg] = and Q = if R [sg] = 0. Laurent series expansion theorem for holomorphic functions. Let the function f(z) be defined and CD (holomorphic) in an open set U which contains as a subset an open circular ring U q,r (z o ), centered at some z o with 0 q < r, including possibly r = : U q,r (z o ) U and 0 q < r. Then f(z) has a unique Laurent series expansion around z o with inner and outer convergence radii, Q and R, obeying 0 Q q < r R, including possibly R =. This means that there exists a unique sequence of complex coefficents, a 0, a 1, a 2,..., and a unique sequence of complex coefficents, b 1, b 2,..., so that f(z) = a k (z z o ) k + k=0 b k (z z o ) k for all z in U q,r (z o ); k=1 and both the foreoging a- and b-series are guaranteed to converge at least for all z in the non-empty open ring U q,r (z o ). 19

20 (C) Isolated Singularities and Residues. Definition of isolated singularities and their residues. A particularly important application of the Laurent expansion involves the case of z o being an isolated singularity of f(z). Assume that V is an open set, z o is a point in V and f(z) is defined and CD (holomorphic) for all z in V, except for z o. That is, f(z) is defined and CD (holomorphic) on an open set U; and U conssist of all points of V, except that the point z o is removed: or, equivalently, U = V \{z o } and z o V ; V = U {z o } and z o / U. We call such a point z o an isolated singularity of f(z). That is, z o is an isolated singularity of f(z) if f(z) is defined and CD (holomorphic) on an open set U, the point z o is not in U, but if z o is added to U, the resulting set V = U {z o } is also still an open set. Assume, for example, that the open set U is the circular disk enclosed by the unit circle around z = 0 with radius 1, but excluding the center z = 0 and excluding the circumference, z = 1. That is, U = U 0,1 (0) in the circular ring notation defined above. If we add to this set U the point z o = 1 on the circumference of U, then the resulting set V = U {1} is not an open set anymore: if we draw a circular disk U r (1) around z o = 1 with any radius r > 0, then that disk U r (1) can never be fully contained within V. No matter how small we make r, some part of U r (1) will lie outside of the unit circle circumference and hence outside of V. Thus, z o = 1 is not a singular point. On the other hand, if we add z o = 0 to U, then the resulting set V = U {0} is just the circular disk V = U 0 (1) and this is still an open set: if we draw a smaller circular disk U r (0) around z o = 0, with radius r = 1/2 > 0 say, then that smaller disk, U 1/2 (0), will be fully contained within V. So, V is still an open set, just like U. So, loosely speaking, an isolated singularity of f(z) is not contained in the function s domain U, but it is completely surrounded by points contained inside U. Since V is open and contains the singularity z o, there must be a circular disk U r (z o ) around z o with some non-zero radius r > 0 which is fully contained within V : and thus U r (z o ) V U r (z o )\{z o } V \{z o } = U. So, if we remove the singular point z o from U r (z o ) the resulting circular disk without its center z o is fully contained in U. But that circular disk without its center is also a circular ring in the sense of the Laurent expansion theorem: U r (z o )\{z o } = U 0,r (z o ) and f(z) is CD (holomorphic) on the open set U containing that ring U 0,r (z o ) as a subset. Hence, we conclude that f(z) can be expanded into a Laurent series around z o and the 20

21 expansion defines a unique sequence of a-coefficients a 0, a 1, a 2,... and a unique sequence of b-coefficients b 1, b 2,... So, f(z) can be expressed as f(z) = a k (z z o ) k + k=0 b k (z z o ) k k=1 for all z with 0 < z z o < r and both the foreoging a- and b-series are guaranteed to converge at least for all z in the non-empty open ring 0 < z z o < r. Of particular interest in this expansion is the coefficient b 1 and because of its importance it has a special name: it is called the residue of f(z) at the singular point z o. We call this process of obtaining the b 1 coefficient from f(z) by Laurent expansion around z o the Res operation and denote its result by Resf(z o ) := b 1. Singularity zoology: removable, pole and essential singularities. If z o is an isolated singularity of a CD (holomorphic) f(z) on an open set U, then z o falls into one of the following three categories: (1) z o is a removable singularity if lim f(z) z z o exists. We can then extend the domain of f(z) to the open set V = U {z o }, by setting f(z o ) := lim z zo f(z). The extended f(z) is then guaranteed to be holomorphic on V ; and all the b-coeffients in the Laurent expansion of f(z) around z o are guaranteed to vanish, b k = 0 for all k = 1, 2, 3,.... The Laurent expansion consists just of the a-series: it is the Taylor series expansion of f(z) around z o. Also, obviously, the residue of f(z) at a removable singularity z o vanishes: Resf(z o ) = 0. Note however that a vanishing residue does it not imply that the singularity is removable: Resf(z o ) only gives the value of b 1, and it is possible to have b 1 = 0 while b k 0 for (some) k 1. (2) z o is a pole singularity or pole, for short, if lim (z z o ) n f(z) exists and is non-zero for some positive integer n 1. z z o The smallest integer n for which lim z zo (z z o ) n f(z) exists is called the order of the pole z o ; and we assume now that n denotes the order of the pole. 21

22 We are then guaranteed that only a finite number of the b-coeffients in the Laurent expansion of f(z) around z o are non-zero and they obey: b n = lim z zo (z z o ) n f(z) 0 and b k = 0 for all k n + 1 ; The b-series in the Laurent expansion around z o thus terminates after a finite number of terms: f(z) = k=0 a k (z z o ) k + b 1 (z z o ) + b 2 (z z o ) b n (z z o ) n for all z with 0 < z z o < r. Evaluating the residue, Resf(z o ) = b 1, can be accomplished by means of one of the Residue Recipes described below, or extensions thereof. (3) z o is an essential singularity if lim (z z o ) n f(z) does not exist for any positive integer, n 1. z z o In this case, we are guaranteed that there is an infinite number of non-zero b-coefficients in the b-series of the Laurent expansion of f(z) around z o, i.e., the b-series does not terminate after any finite number of b k -terms. Evaluating the residue, Resf(z o ), can be difficult in this case. Quartum non datur. 22

23 Residue Recipes (A) If q(z) is complex differentiable on an open set containing z o and h(z) = q(z) z z o then Res h(z o ) = q(z o ). (B) If q(z) is complex differentiable on an open set containing z o and h(z) = q(z) (z z o ) 2 then Res h(z o ) = q (z o ). (C) If q(z) is complex differentiable on an open set containing z o ; and n = 0, 1, 2, 3,... is any non-negative integer; and h(z) = q(z) (z z o ) n+1 then Res h(z o ) = 1 n! q(n) (z o ). (D) If q(z) and D(z) are complex differentiable on an open set containing z o, with and D(z o ) = 0 and D (z o ) d dz D(z) zo 0, h(z) = q(z) D(z) then Res h(z o ) = q(z o) D (z o ). (E) If h(z) is complex differentiable on an open set excluding z o and has an n-th order pole at z o ; and m is an integer m n; and H(z) = (z z o ) m h(z) then Res h(z o ) = 1 d (m 1) (m 1)! dz H(z) zo. m 1 23

24 Complex Line Parameterization and Line Integrals (A) Parameterization To evaluate a complex line integral f(z) dz by explicit integration along C, one must first C find a parameterization, z(τ), for the integration curve C. Here, z(τ) is a complex-values function defined on a real parameter interval [a, b], i.e., for a τ b. The curve C starts at z a := z(a) and ends at z b := z(b). As τ runs from a to b, the complex values z(τ) trace out all points on the curve C. The complex line integral is then expressed as a definite integral over the real variable τ, taken from a to b: b f(z) dz = f(z(τ)) d z(τ) dτ. dτ C a For a curve C consisting of two or more connected straight or circular segments, the integral along C can be broken up into the sum of integrals along each segment and then a parameterization can be applied to integrate over each segment. For any curve C, one can construct infinitely many equivalent parameterizations z(τ). The following describes a few useful possible parameterizations of a straight line segment and of a circular arc segment. (B) Straight Line from z a to z b. z(τ) = z a + (z b z a ) τ a b a with a < b, τ [a, b] The choice of the real parameter interval bounds a and b is arbitrary, as long as a < b. (The reader may want to try different ones to see this explicitly!) A convenient choice is sometimes a = 0 and b = 1; or a = 1 and b = 1. Another convenient choice is very often: a = 0 and b = L where L is the length of the straight line segment. (C) Circular Arc Segment from Angle θ a to Angle θ b on a circle of radius R. Assume the circle of radius R is centered at an arbitrary complex number z o ; and the arc sweeps over a range of angles from θ a to θ b, with all angles measured from the direction of the positive real axis. The arc thus starts at z a = z o + Re iθa and ends at z b = z o + Re iθ b. Then, z(τ) = z o + Re iτ, τ [a, b] with a θ a, b θ b, if θ a < θ b ; or z(τ) = z o + Re iτ, τ [a, b] with a θ a, b θ b, if θ b < θ a. The case θ a < θ b represents a counterclockwise arc with parameter interval bounds a θ a and b θ b ; and the case θ b < θ a represents a a clockwise arc, with parameter interval bounds a θ a and b θ b ; Thus, setting θ a = 0 and θ b = 2π gives a full counterclockwise circle; setting θ a = 0 and θ b = 2π gives a full clockwise circle. Note also that θ a and θ b are not constrained to the standard interval ( π, π] here. So, for example, θ a = π/2 and θ b = 3π/2 is allowed, to parameterize a counter-clockwise semi-circle over a vertical diameter, with its bulge pointing leftward. The reader should make sure to verify all the foregoing statements, 24

25 e.g., by plugging in some numbers for z o, R, θ a and θ b and plotting out the corresponding curves in the complex plane. (D) Upper Bounds for Complex Line Integrals To generate contour integral closures C R h(z) dz along semi-circles C R that vanish in the limit of infinite radius R, one makes use of Upper-Bound Inequalities which are straightforward generalizations of corresponding upper-bound inequalities from real-variable integral calculus. Namely, for any complex integrand function h(z), integrated along any finite-length curve C with parameterization z(τ) defined on the τ-interval a τ b, it can be shown that the following upper-bound inequalities hold: 0 C h(z) dz b where L(C) is the length of the curve C: a L(C) := h(z(τ)) d dτ z(τ) dτ L(C) hmax (C) b a d dτ z(τ) dτ ; and h max (C) is the maximum absolute value of the integrand h(z) along the curve C: h max (C) := max( h(z) ) = max ( h(z(τ)) ). z C τ [a,b] In applying this, for example, to a semi-circle C R of radius R, required for contour integral closures, one can obtain the length of the curve from elementary geometry: L(C R )=πr. So, one doesn t actually have to do the foregoing integral for L(C R ) to get the length of C R. One then only has to prove that [πr h max (C R )] 0 for R. From this follows, by the upper-bound inequality, that lim R C R h(z) dz = 0, thus lim R C R h(z) dz = 0. 25