EEE 203 COMPLEX CALCULUS JANUARY 02, α 1. a t b

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1 Comple Analysis Parametric interval Curve 0 t z(t) ) 0 t α 0 t ( t α z α ( ) t a a t a+α 0 t a α z α 0 t a α z = z(t) γ a t b z = z( t) γ b t a = (γ, γ,..., γ n, γ n ) a t b = ( γ n, γ n,..., γ, γ ) b t a Sum of two trigonometric functions can be obtained by using comple eponentials (phasors). Consider sum of two cosines. Real part of an comple eponential is a cosine. Therefore real part of sum of comple eponentials corresponding to these cosines provide the sum of the two cosines. Acos(θ)+Bcos(θ+β) = Ccos(θ +α) Ae iθ +Be i(θ+β) e iθ( A+Be iβ) A+Be iβ = Ce i(θ+α) = Ce iα e iθ = Ce iα Be iβ Ce iα β α A Page of 3

2 Eample : Find 3cos(θ)+4sin(θ). 3+4e iπ/ = 3 4i = 5e i0.93 3cos(θ)+4sin(θ) = 5cos(θ 0.93) Eample : In the following circuit (upper circuit) AC source; cos(t) is connected for a long time and the responses to this source are at steady state. Since cos(t) = Re [ e it], the voltage on capacitor for AC source [ is real part of the output on the capacitor for the comple eponential source; v o (t) = Re V o e iθ e it] (middle circuit). The voltage-current ratio (impedance) of the R and C for the eponential ecitation are constant and Ω and i 3Ω respectively. Consequently the circuit is reduced to a DC resistance circuit (lower circuit). The circuit is a simple voltage divider and the capacitor voltage is V o e iθ ( = i 3 i ) 3. Using this methodology the steady state voltage v o (t) on the capacitor can be easily obtained. V o e iθ = i ( i 3 ) = i ( ) 3 3 i = ( ) i 3+3 = 3e iπ/6 v o (t) = Re [ V o e iθ e it] = Re [ 3e iπ/6 e it] = Re [ 3e i(t π/6)] = 3cos(t π/6) Volt Page of 3

3 Ω cos(t) + 3 F v o (t) Ω e it + 3 F V o e i(t+θ) = V o e iθ e it Ω + i 3F V o e iθ A list of definitions and theorems from the tetbook Fundamentals of Comple Analysis for Mathematics, Science and Engineering by E. B. Saff and A. D. Snider. Definition: Let f (z) be a function defined in the neighborhood of z 0. Then f (z) is continuous at z 0 if lim f (z) = f (z 0 ). z z 0 Definition: Let f (z) be a comple-valued function defined in the neighborhood of z 0. Then the derivative of f (z) at z 0 is given by f (z 0 ) = lim z 0 f (z 0 + z) f (z), z Page 3 of 3

4 provided this limit eists. Definition: A comple-valued function f (z) is said to be analytic on an open set G if it has a derivative at every point of G. Theorem: Let f (z) = u(,y) + iv(,y) be defined in some open set G. If the first partial derivatives of u and v are continuous and satisfy the Cauchy-Riemann equations at all points of G, then f (z) is analytic in G. Definition: A real-valued function φ(,y) is said to be harmonic in a domain D if all its secondorder partial derivatives are continuous in D and if at each point of D, φ satisfies φ + φ y = 0. Theorem: If f (z) = u(,y) + iv(,y) is analytic in a domain D, then each of the functions u(,y) and v(,y)are harmonic in D. De Moivre s formula [rcos(θ)+irsin(θ)] n = r n cos(nθ)+ir n sin(nθ) Elementary functions. The comple eponential function. Definition: If z = +iy, then e z is defined to be a comple number e z = e (cos(y)+isin(y)). Definition: Given any comple number z, we define sin(z) = eiz e iz, cos(z) = eiz +e iz. i Definition: For any comple number z we define sinh(z) = ez e z, cosh(z) = ez +e z. The logarithmic function. Definition: If z 0, then we define log(z) to any of the infinitely many values log(z) = Log( z )+i(θ +πk), k Z, where θ denotes a particular value of arg(z). The branch of the logarithm for k = 0 is called the principal value of log(z) and we refer it as the principal value of log(z). We denote this function by Log(z), i.e., Log(z) = Log( z )+iarg(z). Definition: If α is a comple constant and z 0, then we define z α by z α = e αlog(z) Page 4 of 3

5 If α is not a real rational number, we obtain infinitely many different values for z α. On the other hand, if α = m/n where m and n > 0 are integers having no common factor, then n distinct values of z m/n, namely z m/n = e (m/n)log( z ) e i(m/n)(arg(z)+πk) (k = 0,,...,n ). Definition: A point set γ in the comple plane is said to be a smooth arc if it is the range of some continuous comple-valued function z = z(t), a t b, which satisfies the following conditions: (i) z(t) has a continuous derivative on [a, b], (ii) z (t) never vanishes on [a, b], (iii) z(t) is one-to-one on [a, b]. A point set γ is called a smooth closed curve if it is the range of continuous function z = z(t), a t b, satisfying conditions (i) and (ii) above and the following: (iii) z(t) is one-to-one on the half-open interval [a, b), but z(b) = z(a) and z (b) = z (a). The phrase γ is a smooth curve means that γ is either a smooth arc or a smooth closed curve. Definition: A contour is either a single point z 0 or a finite sequence of directed smooth curves (γ,γ,...,γ n ) such that the terminal point of γ k coincides with the initial point of γ k+ for each k =,,...,n. is said to be a closed contour or a loop if its initial and terminal points coincide. A simple closed contour is a closed contour with no multiple points other than its initial-terminal point; in other words, if z = z(t), a t b, is a parametrization of the closed contour, then z(t) is one-to-one on the half-open interval [a,b). Theorem: A simple closed contour separates the plane into two domains, each having the contour as its boundary. One of these domains, called the interior, is bounded; the other called the eterior, is unbounded. The direction along can be completely specified by declaring its initial-terminal point and stating which domain (interior or eterior) lies to the left, we say that is positively oriented (counterclockwise direction). Otherwise is said to oriented negatively (clockwise direction). The length of a smooth curve γ : z = z(t), a t b. l(γ) = b a z (t)dt The contour integral. Consider a function f (z) which is defined over a directed smooth curve γ with initial point α and terminal point β. Page 5 of 3

6 For any positive integer n, we define a partition P n of γ to be a finite number of points z 0,z,...,z n on γ such that z 0 = α, z n = β, and z k precedes z k on γ for k =,,...,n. If we compute the arc length along γ between every consecutive pair of points (z k,z k ), the largest of these lengths provide a measure of fitness of the subdivision; the maimum length is called the mesh the partition and is denoted by µ(p n ). Now let c,c,...,c n be any points of γ such that c lies on the arc from z 0 to z, c lies on the arc from z to z, etc. Under these circumstances the sum S(P n ) defined by S(P n ) = f (c )(z z 0 )+f (c )(z z )+ +f (c n )(z n z n ) is called Riemann sum for the function f corresponding to the partition P n. On writing z k z k = z k, this becomes S(P n ) = n f (c k )(z k z k ) = k= n f (c k ) z k. k= Definition: Let f (z) be a comple-valued function defined on the directed smooth curve γ. We say that f (z) is integrable along γ If there eist a comple number L which the limit of every sequence of Riemann sums S(P ),S(P ),...,S(P n ),... corresponding to any sequence of partitions of γ satisfying lim n µ(p n) = 0; i.e., lim S(P n) = L whenever lim µ(p n) = 0. n n The constant L is called integral of f (z) along γ, and we write L = γ f (z)dz or L = γ f. Theorem: If f (z) is continuous on the directed smooth curve γ, then f (z) is integrable along γ. Theorem: If the comple-valued function f (t) is continuous on [a,b] and F (t) = f (t) for all t in [a,b], then b a f (t)dt = F (b) F (a). Theorem: Let f (z) be a function continuous on the directed smooth curve γ. Then if z = z(t), a t b, is any admissible parametrization of γ consistent with its direction, we have b b f (z)dz = f (z(t))z (t)dt = f (z(t)) dz dt (t)dt. γ a a Corollary: If f (z) is continuous on the directed smooth curve γ and if z = z (t), a t b, and z = z (t), c t d, are any two admissible parameterizations of γ consistent with its direction, then b a d f (z (t))z (t)dt = c f (z (t))z (t)dt. Page 6 of 3

7 Definition: Supposethat is a contour consisting of the directed smooth curves (γ,γ,...,γ n ), and let f (z) be a function continuous on. Then the contour integral of f (z) along is denoted by the symbol f (z)dz and is defined by the equation f (z)dz = f (z)dz + f (z)dz + + f (z)dz. γ γ γ n Theorem: If f (z) is continuous on the contour and if f (z) M for all z on, then f (z)dz Ml(). Theorem: Supposethat thefunctionf (z) is continuous in adomain D andhas an antiderivative F (z) throughout D; i.e., df (z)/dz = f (z) at each z in D (F (z) is analytic in D). Then for any contour lying in D, with initial point z I and terminal point z T, we have f (z)dz = F (z T ) F (z I ). Corollary: If f (z) is continuous in a domain D and has an antiderivative F (z) throughout D, then f (z)dz = 0 for all loops lying in D. Definition: A simply connected domain D is a domain having the following property: If is any simple closed contour lying in D, then the domain interior to lies wholly in D. Theorem (Green s Theorem; Curl Theorem in the Plane): Let V = (V,V ) be a continuously differentiable vector field defined on a simply connected domain D, and let be a positively oriented simple closed contour in D. Then the line integral of V around equals the integral of ( V / V / y), integrated with respect to area over the domain D interior to ; i.e., ( V (V d+v dy) = d V ) ddy. dy D The left hand side of this equation is the work done by the force; V = V (,y) + iv (,y) traversing the closed contour which is the boundary of the surface D. Theorem (Cauchy s Integral Theorem): If f (z) is analytic in a simply connected domain D and is any loop (closed contour)in D, then f (z)dz = 0. Theorem: In a simply connected domain, an analytic function has an antiderivative, its contour integrals are independent of path, and its loop integrals vanish. Page 7 of 3

8 Theorem (Cauchy s Integral Formula): Let be a simple closed positively oriented contour. If f (z) is analytic in some simply connected domain D containing and z 0 is any point inside, then f (z 0 )dz = πi f (z) z z 0 dz. Theorem: Let g be continuous on the contour, and for each z not on set g(ξ) G(z)dz ξ z dξ. Then the function G is analytic, and its derivative is given by G g(ξ) (z)dz = (ξ z) dξ for all z not on. Theorem: If f is analytic in a domain D, then all its derivatives f,f,...,f (n),... eist and are analytic in D. Theorem: If f = u+iv is analytic in a domain D, then all partial derivatives of u and v eist and are continuous in D. Theorem (Morera s Theorem): If f (z) is continuous in a domain D and if f (z)dz = 0 for every closed contour in D, then f (z) is analytic in D. Theorem: If f is analytic inside and on the simple closed positively oriented contour and if z is any point inside, then f (n) (z) = n! f (ξ) πi (ξ z) n+dξ (n =,,3,...). Another form of this equation f (z) (z z 0 ) m dz = πif(m ) (z 0 ) (m )! (z 0 inside ). The Cauchy s residue theorem and method for calculating residues are quotes from dcs/courses/math39/lectures/lecture-45.pdf. Theorem (Cauchy s Residue Theorem): Suppose C is a positively oriented, simple closed contour. If f is analytic on and inside C ecept for finite number of singular points z,z,...,z n, then n f (z)dz = πi Res f (z). z=z k C k= Page 8 of 3

9 Method for calculating residues. Let f be a function with a pole of order m at P. Then ( ) Res f (z) = m ((z P) m f (z)) z=p (m )! z z=p Partial fraction decomposition. N (z) D(z) = A l = N (z) (z P) m ( (l )! z = + m l= ) l ( (z P) m N (z) D(z) A l (z P) l + ) z=p (a+b) = a +ab+b (a+b) 3 = a 3 +3a b+3ab +b 3 = z + z y = z z i Page 9 of 3

10 QUESTIONS Q) A simple closed contour; = (γ, γ, γ 3 ), is given. The sequence of the directed smooth curves of this contour are given below. γ : z (t) = t+i ( 3t t ), 0 t γ : z (t) = t+i( t), 0 t γ 3 : z 3 (t) = cos(π(t+))+isin(π(t+)), 0 t Obtain a contour parametrization for by employing the techniques of rescaling; rescale so that γ is traced as t varies between 0 and 3, γ is traced for 3 t 3, and γ 3 is traced for 3 t. y γ γ γ 3 Q) Compute ( e z +e z ) (e z +e z ) + ( e z e z ) (e z e z ) Q Using the result obtained find the trigonometric identity for cosh(z +z ). (a+b)(c+d)+(a b)(c d) = ac+bd. Q3) Compute ( e z +e z ) (e z e z ) + ( e z e z ) (e z +e z ) Using the result obtained find the trigonometric identity for sinh(z z ). (a+b)(c d)+(a b)(c+d) = ac bd. Q4) Evaluate the following contour integral. z +z +3 z 3 dz Q5) In the following circuit AC source; 3cos(t π/) Volt is connected for a long time and the responses to this source are at steady state. obtain the steady state voltage v o (t) on the Page 0 of 3

11 y Q4 Ω 3cos(t π/) + 3 H v o (t) Q5 inductor. Q6) A comple function u(,y) = y is given. Is it is harmonic? If yes, find harmonic conjugate of this function. Q7) A comple function u(,y) = conjugate of this function. is given. Is it is harmonic? If yes, find harmonic +y Q8) The comple logarithm function is defined as in the following. z = re iθ log(z) = Log(r)+θ +πk, k Z A branch of the logarithm is L 0 (z) = Log(r)+θ, for 0 < θ π. What is the domain that L 0 (z) is analytic. Find L 0 ( +i0 + ), and L 0 ( +i0 ). Obtain d dz L 0(z). Page of 3

12 Q9) Two vectors fields; V(,y) = ( y, ), and V(,y) = ( y, y ) are given. For each of the vector field and the following contours test Green s theorem. y y Q0) Find the singularities and the residues of the following comple function. f (z) = (+i)z 3iz + i+ i z i + +i z +i 3 (z +) + z + Q9 Q) Compute the following integral along the simple closed contour traversed once counter clockwise direction. cos(z) z(z +i) dz y Q Q) Evaluate the following integral along the simple closed contour traversed once counter clockwise direction. e z sin (z) z(z +4) dz Page of 3

13 y Q Page 3 of 3