Lecture_3.pdf - ENGG 2420C Complex Analysis and...

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Unformatted text preview: ENGG 2420C Complex Analysis and Differential Equations for Engineers Prof. Mayank Bakshi e-mail: [email protected] September 10, 2018 Prof. Mayank Bakshi ENGG 2420C 1/1 Overview So far Representing complex numbers: z = Re(z) + i Im(z) = |z| (cos(Arg(z)) + i sin(Arg(z))) Polar form = |z|ei Arg(z) Euler’s Theorem = |z|ei Arg(z)+i2πk for all k = 0, ±1, ±2, . . . since ei2πk =1 (verify!) Operations with complex numbers: z1 + z 2 , z1 z2 , z (Integer) powers of z. z n = |z|ei Arg(z) n = |z|n ei n Arg(z) Note: Arg(z n ) may not equal n Arg(z) in general. To get Arg)(z n ) from n Arg(z), add or subtract integer multiples of 2π until you get something n the range (−π, π]. Today: Roots of complex numbers More examples of complex functions: trigonometric functions, hyperbolic functions, logarithm, power function Prof. Mayank Bakshi ENGG 2420C 3/1 Exercises Compute the modulus, phase, real part and imaginary part of the following complex numbers z 3 , if z = 1 + i z −1 , if z = 1 + 2i z1 /z2 , if z1 = 1 + 2i and z2 = 1 + i fourth root of −1 square root of z1 z¯2 where z1 = 1 + 2i and z2 = 1 + i Prof. Mayank Bakshi ENGG 2420C 20/1 Trigonometric functions Based on Euler’s formula, the trigonometric functions cos and sin are extended to analytic functions in the complex plane by eiz + e−iz eiz − e−iz cos z = and sin z = 2 2i Same properties as the real trigonometric functions (cos z)0 = − sin z and (sin z)0 = cos z cos(z1 + z2 ) = cos z1 cos z2 − sin z1 sin z2 sin(z1 + z2 ) = sin z1 cos z2 + cos z1 sin z2 cos2 z + sin2 z = 1 etc. NOTE: Matlab function cos, sin. Prof. Mayank Bakshi ENGG 2420C 21/1 Hyperbolic functions The hyperbolic functions are defined from the complex exponential by ez + e−z ez − e−z cosh z = and sinh z = 2 2 Properties derived from their link with trigonometric functions: cosh z = cos iz and sinh z = −i sin iz (cosh z)0 = sinh z (sinh z)0 = cosh z cosh2 z − sinh2 z = 1 etc. NOTE: Matlab function cosh, sinh. Prof. Mayank Bakshi ENGG 2420C 22/1 Complex logarithm We say that a = ln z if ea = z. This definition extends the definition of real natural logarithms. Note that z = |z|ei arg z = eln |z|+i arg z = eln |z|+i arg z+i2nπ for any integer n. Therefore, we define the multi-valued function ln z as ln z = ln |z| + i arg z + i2nπ, n = 0, ±1, ±2, . . . . When n = 0, the corresponding value of the logarithm is called the Principal Value. Definition  The Principal Value of the complex “natural” logarithm is defined by Ln z = ln |z| + i arg z Prof. Mayank Bakshi ENGG 2420C 23/1 Properties 1 −π < Im(Ln z) ≤ π (Why?); 2 ln z = Ln z + i2nπ, n = 0, ±1, ±2, . . .; 3 5 eln z = z;  ln ez = z + i2nπ, n = 0, ±1, ±2, . . .;  Ln ez = z + i2kπ for an appropriate k † ; 6 ln(z1 z2 ) = ln z1 + ln z2 + i2nπ, where n is an integer† ; 7 ln is analytic in C\] − ∞, 0]; 8 (ln z)0 = 1/z. 4 †k is chosen uniquely, so that the imaginary part of the rhs is > −π and ≤ π. NOTE: Matlab function log. Prof. Mayank Bakshi ENGG 2420C 24/1 Selected proofs: 3  eln z = eln |z|+i arg z = eln |z| cos(arg z) + i sin(argz) = |z| cos(arg z) + i sin(arg z) = z 7 Let ln z = u + iv for z ∈ C\] − ∞, 0], then from ln z 1 we have ln z ∂{e } ∂{e } and i = ∂x ∂y = (ux + ivx )eu+iv = (uy + ivy )eu+iv = (ux + ivx )z = (uy + ivy )z 1= Hence i(ux + ivx ) = (uy + ivy ) (Cauchy-Riemann), which proves analyticity. 8 from 4 we have that ux + ivx = 1/z. On the other hand, (ln z)0 = ux + ivx , hence the result. Prof. Mayank Bakshi ENGG 2420C 25/1 Power function The arbitrary power of a complex number is defined by z a = ea ln z . Again, this is a multi valued function as ln z is multi valued. Definition  The principal value of an arbitrary power of a complex number is z a = ea Ln z Properties 1 Ln z a = a Ln z + 2kiπ, for an appropriate integer k † ; 2 z a z b = z a+b ; 3 z a is analytic in C\] − ∞, 0]; 4 (z a )0 = az a−1 . NOTE: in general, z a †k b 6= z ab . chosen uniquely, so that the imaginary part of the rhs is > −π and ≤ π. Prof. Mayank Bakshi ENGG 2420C 26/1 Exercises Give the polar and cartesian expressions of √ def x + iy = (x + iy)1/2 in function of x and y ln(1 + i) ln(−1) ln(z1 z1 ), if z1 = −1 + i and z2 = −1 + 2i sin(π + i) Prof. Mayank Bakshi ENGG 2420C 27/1 Calculus using MatLab  MatLab is a widely spread software for computing with complex numbers in double precision† . No compilation needed. Example: Computation and plot of the derivative of f (x) = e−x 2 1 x = -2:0.001:2; diff_x = 1e-6; diff_f = exp(-(x+diff_x).^2)-exp(-x.^2); deriv_f = diff_f/diff_x; plot(x,deriv_f) 0.5 0 −0.5 −1 −2 0 −1 0 −x2 The difference with the exact derivative f (x) = −2xe 1 2 is given by max(abs(-2*x.*exp(-x.^2)-deriv_f)) ans = 1.00007824956408e-06 MatLab is particularly useful for checking numerically the result of mathematical computations. † Relative precision: 2−52 ≈ 2.10−16 Prof. Mayank Bakshi ENGG 2420C 31/1 Definition of complex functions A complex function, f , is a function of the complex variable z = x + iy that results in a complex-valued output f (z) = u(x, y) + iv(x, y) where u(x, y) and v(x, y) are real functions of two variables. A complex function is continuous at a point z0 iff, for any neighborhood† V of f (z0 ), f −1 (V) is a neighborhood of z0 (f −1 (V) is the set of points z for which f (z) ∈ V). f (z0 ) x z x z0 x neighborhood of z0 † e.g., f (z) x neighborhood of f (z0 ) a disk centered at f (z0 ) with non-zero radius. Prof. Mayank Bakshi ENGG 2420C 32/1 Continuity of complex functions A complex function, f (z) = u(x, y) + iv(x, y) is continuous at a point z0 iff, for any neighborhood† V of f (z0 ), f −1 (V) is a neighborhood of z0 (f −1 (V) is the set of points z for which f (z) ∈ V). f (z0 ) x z x z0 x neighborhood of z0 f (z) x neighborhood of f (z0 ) NOTE: To show that f (z) is continuous, once can show that u(x, y) and v(x, y) are both continuous functions of x and y. e.g. f (z) = y − ix2 is continuous since both u(x, y) = y and v(x, y) = −x2 are continuous functions of x and y. † e.g., a disk centered at f (z0 ) with non-zero radius. Prof. Mayank Bakshi ENGG 2420C 33/1 Announcements and Reminders ; Lecture tomorrow 10:30-11:15 in ERB 407. Roll call may be taken. ; Tutorial from 11:15-12:15. Find your tutorial sections on piazza. Quiz 1 on Sep 20 in tutorial. Syllabus: everything upto lecture on Monday, Sep 17. Prof. Mayank Bakshi ENGG 2420C 34/1 ...
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