Complex Numbers PDF

# Complex Number - Equation like 171=0, 22-10x+40=0 do not have real solution and this leads to the introduction of complex numbers. - A complex number (z) is a ordered pair of real numbers (x, y), where x is the real part of z and y is called the imaginary part of z, this written as z = x + iy or...

# Complex Number - Equation like 171=0, 22-10x+40=0 do not have real solution and this leads to the introduction of complex numbers. - A complex number (z) is a ordered pair of real numbers (x, y), where x is the real part of z and y is called the imaginary part of z, this written as z = x + iy or z = (x, y). - Equality of two complex numbers z1 = x1 + iy1, z2 = x2 + iy2. They are equal only if and only if their real and imaginary parts are equal. - Imaginary unit (i) - 0,1 is called Imaginary unit (i) - (i) = (0,1) - Algebra of Complex Number - If z = x1 + iy1, & z2 = x2 + iy2, then their sum, product, difference and quotient are calculated as following: - Sum = z1 + z2 = (x1 + x2) + i(y1 + y2) - Product: z1 * z2 = (x1 + iy1)(x2 + iy2) = x1x2 - y1y2 + i(x1y2 + x2y1) - Difference z1 - z2 = (x1 - x2) + i(y1 - y2) - Quotient: z1/z2 = (x1x2 + y1y2) / (x2^2 + y2^2) + i(x2y1 - x1y2) / (x2^2 + y2^2) - Find the sum, difference, quotient and product of z1 = 8 + 5i and z2 = 9 - 2i. - Sum: z1 + z2 = 8 + 5i + 9 - 2i = 17 + 3i - Difference: z1 - z2 = 8 + 5i - (9 - 2i) = -1 + 7i - Product: z1 * z2 = (8 + 5i)(9 - 2i) = 72 + 6 + i(27 - 16) = 78 + i(11) - Quotient: z1/z2 = (8 + 5i) / (9 - 2i) = (72 + 6) / (81 + 4) + i(27 - 16) / (81 + 4) = 78 / 85 + i(11) / 85 - Geometrical representation of a complex number: We know that every complex number z is an ordered pair of real numbers(x, y) where x is the real part of z and y is the imaginary part of z. Denoting x as real axis and denoting y as imaginary axis in the Cartesian form, we plot the point P with coordinates (x, y), this is called the complex plane. - Complex conjugate number: Complex conjugate of a complex number is given by z = x + iy, z = x - iy. This conjugate of complex number is its reflection about its xx axis, for example, z = 5 + 2i, z = 5 - 2i. - Multiplication of z by i is geometrically a counterclockwise rotation by 90 degrees, verify that by plotting z and iz, the angle of rotation is 90 degrees. - Polar form of complex number: r and $\theta$ are the polar coordinates of a point in the Cartesian plane. - By the relations x = r cos$\theta$, y = r sin$\theta$, the polar form of a complex number is: z = x + iy = r cos$\theta$ + i r sin$\theta$ = re^(i$\theta$). - Polar form = z = re^(i$\theta$) - Cartesian form = z = x + iy = r (cos$\theta$ + i sin$\theta$) - Where r is called the absolute value or modulus of z. - |z| = $\sqrt(x^2 + y^2)$. - Geometrically, |z| is the distance of the point z from the origin. - $\theta$ is called the argument of z (argz). - argz = $\theta$ = tan^(-1)(y/x). - Geometrically, $\theta$ represents the directed angle from positive x axis to OP. - **Circles & Discs** - |z - z0| = r represents a circle of radius r, centered at (x0, y0). - |z - z0| < r represents an open circular disc. - |z - z0| > r represents an exterior of a circle. - |z - z0| = 0 represents a point at the origin. - |z - z0|^2 = r^2 represents a circle centered at the origin. - |z - z0| ^2 < r^2 represents the region between two circles, called as annulus region. - |z - z0| ^2 > r^2 represents the region exterior to the closed circler disc. - |z - z0| = r represents a circle of radius r, centered at (xo, y0), Eg: z = x + iy, z0 = xo + i yo, - |z - z0| = r => |x + iy - xo - i yo| = r => $\sqrt((x - xo)^2 + (y - yo)^2)$ = r - |z| = 1 represents a circle centered at the origin with radius 1 unit. - |z| > r represents exterior of the circle only, example of a circle is: |z| = 1 => $\sqrt(x^2 + y^2)$ = 1 => x^2 + y^2 = 1. - |z - z0| < r represents the interior of the circle only. - |z - z0| ≤ r represents the interior & boundary of the circle. - **Half planes:** - *Set of all points of the form z = x + iy such that, y = 0 is the upper half plane. - *Set of all points, z = x + iy such that, y > 0 is the upper half plane. - *Set of all points, z = x + iy such that y < 0 is the lower half plane. - *Set of all points, z = x + iy such that x ≥ 0 is called right half plane - *Set of all points, z = x + iy such that x ≤ 0 is called left half plane. - **Determine and plot/sketch in complete plane, the following sets:** - 1) |z + (2 + 5i)| = 4 => $\sqrt(x + 2)^2 + (y + 5)^2 = 4$ - 2) Imz ≤ π => -π ≤ y ≤ π. - 3) -$\sqrt(2)$ < |z - 1 - i| < $\sqrt(2)$ => 2 <= (x - 1)^2 + (y - 1)^2 <= 2 - 4) Re(z) < 1 => x < 1. - **Complex variable, complex function** - z = x + iy - f(z) = u + iv => complex function - f(z) = u(x, y) + i v(x, y) - A function f defined on a set S of complex numbers is a rule that assigns to every z in the set S, a complex number (w) is called as complex variable - f : S -> (S) - Where z varies in sets S and w is called a complex function, for example w = f(z) = z^2 + 3z, defined for all z i.e., a complex function, find w and calculate the value of w at z = 1 + 3i - w = f(z) = u + iv - u + iv = (x + iy)^2 + 3(x + iy) - u + iv = x^2 - y^2 + i 2xy + 3x + i 3y - u = x^2 - y^2 + 3x, v = 2xy + 3y - **Limits, continuity, and differentiability of a complex function**: - A function f(z) is said to have a limit L as z approaches to z0, written as lim(z->z0) f(z). - A function f(z) is said to be continuous at z = z0 if f(z0) is defined and lim(z->z0) f(z) = f(z0). - A derivative of a complex function of z is written as f'(z0) = lim(Δz->0) [f(z0 + Δz) - f(z0)] / Δz - **Analytic functions (regular functions):** - The functions that are differentiable at some point of their domain are called analytic functions. - A function f(z) is said to be analytic in a domain D if f(z) is well-defined and differentiable at all points of the domain D, for example, non-negative integer power, 1,2,27, 29..... f(z) = co + c1z + c2z^2 + ... + cnz^n, where co, c1, c2..... are (real or complex) are defined as analytic, f(z) = z^2 is analytic, f(z) = 1/z is not analytic, and so on. - **Cauchy-Riemann (C.R) equations:** - A complex function f(z) is said to be analytic in a domain D if and only if the following equations are satisfied: - w = f(z) = u(x, y) + iv(x, y) - ux = vy, uy = -vx - **Verify that f(z) = z^2 is analytic?** - f(z) = u + iv - u + iv = (x + iy)^2 - u +iv = x^2 - y^2 + i 2xy - u = u(x, y) = x^2 - y^2 - ux = 2x, vy = 2x - uy = -2y, vx = -2y - Therefore, ux = vy, uy = -vx. Hence the function is analytic. - **Verify that f(z) = |z|^2 is analytic?** - u + iv = (√(x^2 + y^2))^2 = x^2 + y^2 - u = u(x, y) = x ^ 2 + y^2 - ux = 2x, vy = 2y - uy = 2y, vx = 2x - Hence the function is not analytic, as uy ≠ -vx - **Verify that f(z) = z^3 is analytic?** - u + iv = (x + iy)^3 = x^3 + i3x^2y - 3xy^2 + iy^3 - u = x^3 - 3xy^2 - v = 3x^2y - y^3 - ux = 3x^2 - 3y^2, vy = 3x^2 - 3y^2 - uy = 6xy, vx = 6xy - Hence the function is analytic. - **Verify that f(z) = |z|^2 is analytic?** - u + iv = (√(x ^ 2 + y^2))^2 = x ^2 + y ^ 2 - u = u(x, y) = x ^2 + y^2 - ux = 2x, vy = 2y - uy = 2y, vx = 2x - Therefore, uy ≠ -vx, so the function is not analytic at every point in the domain. - **Verify that f(z) = ln|z| + i arg z is analytic?** - u + iv = ln (√(x^2 + y^2)) + i tan^(-1)(y/x) - u = ln (√(x^2 + y^2)) = 1/2 * ln(x^2 + y^2) - v = tan^(-1)(y/x) - ux = x / (x^2 + y^2), vy = x / (x^2 + y^2) - uy = 1 / (1 + (y/x)^2) * (1/x) - 1 / (1 + (y/x)^2) * (y/x^2) = 1 / (1 + (y^2/x^2)) * (1/x) - y / (x^2 + y^2) * (1/x) = x / (x^2 + y^2), vx = -y / (x^2 + y^2) - Therefore, ux = vy, uy = -vx. Hence the function is analytic. - **Polar form of C.R. equations** - In polar form z = r(cos$\theta$ + i sin$\theta$) = re^(i$\theta$), - w = f(z) = u(r, $\theta$) + iv(r, $\theta$) - Then the c.r. equations are: - ur = 1/r * v$\theta$, v$\theta$ = -1/r * ur - **Verify that f(z) = z^2 is analytic:** - u + iv = (re^(i$\theta$))^2 = r^2 e^(i 2$\theta$) = r^2 (cos 2$\theta$ + i sin 2$\theta$) - u = u(r,$\theta$) = r^2 cos 2$\theta$ - v = v(r,$\theta$) = r^2 sin 2$\theta$ - ur = 2r cos 2$\theta$ - v$\theta$ = 2r^2 cos2$\theta$ - Therefore, ur = 1/r v$\theta$, v$\theta$ = -1/r * ur. Hence the function is analytic - **Derivation of C.R. equations:** - Let f(z) = u(x, y) + iv(x, y) be defined and continuous in some neighborhood of a point z = x + iy, and differentiable at z itself. Then at that point partial derivatives of u and v exists and satisfy the C.R equation. - Given that f’(z) exists, - f’(z) = lim(Δz->0) [f(z + Δz) - f(z)] / Δz exists, - This limit approaches zero along any path in the neighborhood of z. Therefore, we may choose the two paths I and II in fig. 3.35 and equate the results. By comparing the real parts, we obtain the first C.R equation and by comparing the imaginary parts we obtain the second C.R. equation. - Let Δz = Δx + iΔy - Then z + Δz = x + Δx + i(y + Δy) - For path I, where Δy = 0, Δz = Δx - For path II, where Δx = 0, Δz = i Δy - f’(z) = lim(Δx->0) [u(x + Δx, y) -u(x, y) + i lim(Δx->0) [v(x + Δx, y) - v(x, y)] / Δx = ux + ivx - f’(z) = lim(Δy->0) [u(x, y + Δy) -u(x, y) + i lim(Δy->0) [v(x, y + Δy) - v(x, y)] / iΔy = -uy + vy - Therefore, ux + ivx = -uy + vy, hence proved that ux = vy, uy = -vx - **Harmonic functions** - If Laplace’s equation (Δu = 0) is satisfied for u and v in a domain D, then u and v satisfy the Laplace equation in domain D. - A function satisfying the Laplace equation is called harmonic function. - U and v are two harmonic functions. - If two harmonic functions u and v satisfy the C-R equations in domain D, they are called the real and imaginary parts of a function f in domain D. - Vice-versa. - **Verify that u = x^2 - y^2 - y is harmonic in the whole complex plane and construct the analytic function f(z)? Hence find the harmonic conjugate of u.** - u = x^2 - y^2 - y - ux = 2x - uy = -2y - uxx = 2 - uyy = -2 - Therefore, uxx + uyy = 2 - 2 = 0. Hence u is harmonic. - Consider f’(z) = e^(ix) = cosx + i sinx, then f(z) = ∫(e^(ix) dx) = sinx + ic (c is real). - Apply the Thomson's method. - Put x = z, y = 0. Then f(z) = z + iz + c. - Integrate f(z) = z ^ 2 / 2 + i z + c - F(z) = (x + iy) ^ 2 / 2 + i (x + iy) + c = (x^2 - y^2 / 2 + i xy + ix + iy + c = (x ^2 - y^2 / 2 + ix + c) + i (xy + y). - Hence f(z) = (x^2 - y^2) /2 + ix + c) + i (xy + y), where c is a real constant, is the analytic function whose real part is u = x^2 - y^2. - Therefore, the harmonic conjugate of u is v = xy + y. - **Construct the analytic function f(z) = u + iv, whose real part is u = (x^2 + y^2)/(x^2 + y^2)^2** - ux = (x^2 + y^2)^2 * 2x - (x^2 + y^2) * 4x(x ^ 2 + y ^2) / (x^2 + y^2) ^ 4 = 2x(x^2 + y^2) - 4x(x^2 + y^2) / (x^2 + y^2)^3 = - 2x(x^2 + y ^2)/ (x^2 + y^2)^3 = - 2x / (x^2 + y^2)^2 - uy = 2y(x^2 + y^2) - 4y(x^2 + y^2) / (x^2 + y^2)^4 = - 2y(x^2 + y^2) / (x^2 + y^2)^3 = -2y / (x^2 + y^2)^2 - Consider f’(z) = 1 / (z * i * z) = 1 / (i * (x^2 + y^2)). - Apply the Thomson method, where x = z, y= 0. - f(z) = -i / z - Integrate f(z) = - i ln(z) - Therefore, f(z) = -i ln(z) + c, where c is a real constant, is the analytic function whose real part is u = (x ^2 + y^2) / (x^2 + y^2)^2 - **Conformal mapping** - In w = f(z), putting z = x + iy, f(z) separated into real and imaginary parts, w = f(z) can be represented in two separate planes, namely the z plane corresponds to a point (u, v) in the w plane; and the set of points (x, y) traces a curve C in the z plane and the corresponding points (u, v) traces a curve C’ in the w plane. - We say that C is mapped or transformed onto the curve C' under the transformations w = f(z). - The mapping defined by analytic function f(z) is conformal, except at critical points (i.e.) the points at which derivatives of f’(z) = 0. - **Linear fractional transformation (Bilinear transformation)** - *Invariant points (fixed points)*: - If w = z maps onto itself, i.e., if w = z under the bilinear transformation, then the point is called an invariant point or fixed point. - *Theorem 2.* Three given distinct points z1, z2, z3 can always be mapped on to three prescribed distinct points w1, w2, w3 (in w plane) by one and only mapping, linear fractional transformation w = f(z). This mapping is given implicitly by the equation. - (w - w1)(w2 - w3) / (w - w3)(w2 - w1) = (z - z1)(z2 - z3) / (z - z3)(z2 -z1) - The transformation w = (az + b) / (cz + d), where a, b, c, d are real or complex constants, ad - bc ≠ 0 is called linear fractional transformation (bilinear transformation). - **Find the LFT that maps three given points (z1, z2, z3) onto three given points(w1, w2, w3).** - Let z1 = -1, z2 = 0, z3 = 1, w1 = -i, w2 = -1, w3 = 1. Then, - (w - w1)(w2 - w3) / (w - w3)(w2 - w1) = (z - z1)(z2 - z3) / (z - z3)(z2 - z1) => (w + i)(-1 - 1) / (w - 1)(-1 + i) = (z + 1)(0 - 1) / (z - 1)(0 + 1) => (w + i)(-2) / (w - 1)(-1 + i) = -(z + 1) / (z - 1) => (w + i)(-2)(z - 1) = (w - 1)(-1 + i)(z + 1) => (2w + 2i)(-1 + z) = (w - 1)(-iz - i + 2 + 1) => (-2w + 2i)z + (2w - 2i) = -iwz + iz -iw – i + 2w + 2 => -2wz - 2i + 2wz - 2i =-iwz + iz - iw -i + 2w + 2 => (-2w + i)z = -iwz + 1 - i + 2w + 2 => -w(2 + i) + i(-2 + i) = 2 - 1z + 1 => w(-2 - i + zi) = z - iiz + 1 => w (-2 - 1 + i(2 + 1) = z - 1 + 1(2 - 1)=> w = (z - 1 + i(2 - 1))/(-2 - 1 + i(2 + 1)) => w = (z - 1 + i(2 - 1)) * (-(2 + 1) - i(2 + 1)) / ((-2 - 1) + i(2 + 1))((-(2 + 1)) - i(2 + 1)) => w = (-2 - 1)((z - 1) + i(2 - 1)) + i(2 + 1)((z - 1) + i(2 - 1)) / ((-2 - 1)^2 + (2 + 1)^2) => w = (-2 - 1)(z - 1) -i(2 + 1)(2 - 1) + i(2 + 1)(z - 1) + (2 + 1) i(2 - 1) / [(2 + 1)^2 + (2 + 1)^2] => w = (-2 - 1)(z - 1) -i(2 + 1)(z - 1) + i(2 + 1)(z - 1) + (2 + 1) i(2 - 1) / [2(2 + 1)^2] => w = w(z - 1) + i(2 + 1)(z - 1) / [2(2 + 1)^2] => w = (z - 1) + i(2 + 1)(z - 1) / 2(2 + 1) => w = (z - 1) + i(z - 1) / 2 => w = (2z - 2 + iz - i)/ 2 = (2 + i)z - (2 + i)/2 => w = (2 + i)z - (2 + i) / 2 - **Find the invariant points of the transformation w = (z + 1) / (z - 1)** - For a fixed point w = z, z = (z + 1)/(z - 1) => z(z - 1) = z + 1 => z^2 - z = z + 1 => z^2 - 2z - 1 = 0 => z = (2 ± √(2^2 + 4(1)(1)))/2 = (2 ± √(8))/2 = (2 ± 2√(2))/2 = 1 ± √(2) => z1 = 1 + √(2), z2 = 1 - √(2). Therefore, 1 + √(2), 1 - √(2) are two invariant points of the transformation. - **Find the LFT/BLT that maps the points z1 = 1, z2 = i, z3 = -1 onto the points w1 = 1, w2 = 0, w3 = -i respectively. Under this transformation, find the image of |z|≤ 1 and find the invariant point of the transformation.** - (w – w1)(w2 – w3) / (w – w3)(w2 – w1) = (z – z1)(z2 – z3) / (z – z3)(z2 – z1) => (w – 1)(0 + i) / (w + i)(0 – 1) = (z – 1)(i + 1) / (z + 1)(i – 1) - We have a complex value in the numerator and the denominator. Divide the numerator and the denominator by the common term (i + 1). Then the equation becomes (w – 1)(-i) / (w + i)i = (z – 1) / (z + 1) * (i – 1) / (i – 1) => (w + i)(i) / (w + i) = (z – 1)(i – 1) / (z + 1) => (w + i) = (z – 1)(i – 1) / (z + 1) => w + i = (z – 1)i - z + 1 / (z + 1) => (w + i)(z + 1) = i(z - 1) - z + 1 => wz + w + iz + i = iz - i - z + 1 => wz + w + i(z + 1) = 1 - iz - z + 1 => wz + w + i(z + 1) = 2 – i(z + 1) => w(z + 1) = 2 - i(z + 1) - i(z + 1) => w(z + 1) = 2 – 2i(z + 1) => w = (2 – 2i(z + 1)) / (z + 1) - The image of |z| ≤ 1 is calculated in the w-plane by finding the image of a circle in the z-plane. - To find the invariant point, we can use the equation derived in the previous point. w = (2 – 2i(z + 1)) / (z + 1). We can solve this equation for z and simplify as follows: (2 - 2i(z + 1)) / (z + 1) = z => 2 - 2i(z + 1) = z(z + 1) => 2 - 2iz - 2i = z^2 z => 2 - 2i(z + 1) = z^2 + z => 2 - 2iz - 2i = z^2 + z => z^2 + (1 + 2i)z + (2i - 2)= 0. - We can now use the quadratic formula to find the roots of this equation: z = (-(1 + 2i) ± √((1 + 2i)^2 - 4(1)(2i - 2))) / (2(1)). - Using the quadratic formula and simplifying, we get the invariant points as: z = -i, z = -1 - 2i - **Find the transformation, that maps (0, 1, i) onto (1, 0, 2i) and verify the correctness of the theorem. The transformation is:** - (w – w1)(w2 – w3) / (w – w3)(w2 – w1) = (z – z1)(z2 – z3) / (z – z3)(z2 – z1) - (w – 1)(0 – 2i) / (w – 2i)(0 – 1) = (z – 0)(1 – i) / (z – i)(1 – 0) => (w – 1)(-2i) / (w – 2i)(-1) = z(1 – i) / (z – i) => (w –1)(2i) / (w – 2i) = (z - i)(z – 1) => w = (z – i)(z – 1) / (2i(z – 2i)) = (z^2 – iz – z + i) / (2iz - 4i^2) - The denominator can be simplified as: 2iz - 4i^2 = 2iz +4. - (z^2 – iz – z + i) / (2iz + 4) = 1/2 * (z^2 – iz – z + i) / (iz + 2) = (1/2) * (z^2 – iz – z + i) / (iz + 2) * (iz - 2) / (iz - 2) = (1/2){iz (z^2 – iz – z + i) - 2(z^2 – iz – z + i)} / (i^2 * z^2 + 2^2) = 1/2 * (i^2 * z^3 – i^2 * z^2 – iz^2 + iz – 2z^2 + 2iz + 2z - 2i) / (-z^2 + 4) = 1/2 * (-z^3 + z^2 + 2iz + 2z - 2i) / (-z^2 + 4) = 1/2 * (z^3 – z^2 – 2iz – 2z + 2i) / (z^2 - 4). - Therefore, the transformation is: w = 1/2 * (z^3 – z^2 – 2iz – 2z + 2i) / (z^2 - 4). - **Find the transformation that maps the points z1 = 1, z2 = i, z3 = 0 onto the points w1 = 1, w2 = ∞, w3 = -i respectively** - (w – w1)(w2 – w3) / (w – w3)(w2 – w1) = (z – z1)(z2 – z3) / (z – z3)(z2 – z1) - (w - 1)(∞ + i) / (w + i)(∞ – 1) = (z - 1)(i - 0) / (z - 0)(i - 1) => (w - 1)(i) / (w + i)(1) = (z - 1)(i) / z(i - 1) => (w - 1)i / (w + i) = (z - 1)i / z(i - 1) => (w - 1)i * z(i - 1) = (z - 1)i(w + i) => (wi - w)(zi - z) = (z – 1)(wi + w) => (wi - w)(zi - z) = wz – w + zi – i => (wi - w)zi + (w - wi)z + wz – w = 0 => - w(z * i - z + z - 1) + i(z * i - z) = 0 => -w(zi - 1) + i( - z^2 – z) = 0 => -w(zi - 1) - i(z^2 + z) = 0 => -w(zi - 1) - i(z)(z + 1) = 0 => -w(zi - 1) - iz(z + 1) = 0. - This equation can be simplified as: -w(zi - 1) - iz(z + 1) = 0 => -w(zi - 1) - i(z)(z + 1) = 0 => -w(zi - 1) - iz(z + 1) = 0 => -w(zi - 1) - iz(z + 1) = 0. - Therefore, the transformation is: w = (w - 1)i / (w + i) = (z - 1)i / z(i - 1) = (-iz - i) / (z(i - 1)) = (i(z + 1)) / (z(i - 1)) = (i(z + 1)) / (z(i - 1)) = (i + 1) / (z(i - 1))