NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 16
Complex Analysis Module: 2: Functions of a Complex Variable Lecture: 6: Singular points and Applications to problem of potential flow A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 2 / 16
Functions of a complex variable Analytic functions and singular points A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 3 / 16
Analytic functions Singular points Definition Let f (z) fails to be analytic at the point z 0, but analytic at some point in every neighbourhood of z 0 then z 0 is said to be the singular point of f (z). A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 4 / 16
Analytic functions Singular points Example The function f (z) = 1/z fails to be analytic at z = 0. But in every point other than z = 0, the function is analytic. Hence z = 0 is singular point of the function f (z). Further, in other points of the finite complex plane C\{0}, f (z) is analytic and possesses the derivative f (z) = 1/z 2, z 0. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 5 / 16
Analytic functions Singular points Example The function f (z) = z is not analytic at any point z 0 C. In every neighbourhood of z 0 also it is not analytic. Hence f (z) = z is nowhere analytic. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 6 / 16
Analytic functions Entire function Definition A function which is analytic in the entire complex plane is called an entire function. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 7 / 16
Analytic functions Entire function Definition A function which is analytic in the entire complex plane is called an entire function. Example Every polynomial P n (z) of finite degree is an entire function. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 7 / 16
Analytic functions Note Further examples of entire functions will be given later. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 8 / 16
Analytic functions Note Further examples of entire functions will be given later. There are various types of singular points. They are mainly classified as isolated singular points and non-isolated singular points. Further classifications and related discussions will be given later. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 8 / 16
Functions of a complex variable Applications to potential flow A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 9 / 16
Applications Potential Flow Suppose that a fluid flow over the complex plane C and that the velocity at the point z = x + iy be given by the velocity vector V(x, y) = u(x, y) + iv(x, y). (1) A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 10 / 16
Applications Potential Flow Suppose that a fluid flow over the complex plane C and that the velocity at the point z = x + iy be given by the velocity vector V(x, y) = u(x, y) + iv(x, y). (1) We assume that the velocity does not depend on time and u(x, y) and v(x, y) have continuous first order partial derivatives. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 10 / 16
Applications Potential Flow Suppose that a fluid flow over the complex plane C and that the velocity at the point z = x + iy be given by the velocity vector V(x, y) = u(x, y) + iv(x, y). (1) We assume that the velocity does not depend on time and u(x, y) and v(x, y) have continuous first order partial derivatives. The divergence vector field of V is given by divv = u x (x, y) + v y (x, y) and is a measure to the extent to which the velocity diverges near the point. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 10 / 16
Applications Potential Flow We only consider fluid flows that have divergence zero. Such fluids are called irrotational fluid flows. This means that the flow of fluid through any simple closed contour is identically zero. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 11 / 16
Applications Potential Flow We only consider fluid flows that have divergence zero. Such fluids are called irrotational fluid flows. This means that the flow of fluid through any simple closed contour is identically zero. The equation u x (x, y) + v y (x, y) = 0 gives the equation of continuity of the flow. If u x and v y satisfy the Cauchy Riemann equations then by the continuity of partial derivatives we have the corresponding function f = u + iv as analytic function. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 11 / 16
Applications y (x, y + y) (x + x, y + y) (x + x, y) x Figure : 2-D fluid flow A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 12 / 16
Applications Potential Flow Now this analytic function has an antiderivative F(z) such that f (z) = F (z), where F(z) = φ(x, y) + iψ(x, y). This F(z) is called complex potential and has the property that F (z) = φ x (x, y) ψ x (x, y) = u + iv = V(x, y). Thus φ x = u and φ y = ψ x = v. Also gradφ = u + iv = V(x, y). A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 13 / 16
Applications Potential Flow (i) φ(x, y) is called velocity potential and the curves φ(x, y) c 1 (constant) are called Equipotentials of the fluid flow. (ii) ψ(x, y) is called stream line and the curves ψ(x, y) c 2 (constant) are called Stream lines of the fluid flow. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 14 / 16
Applications Potential Flow Example Let the complex potential of the fluid be given by F(z) = (α + iβ)z. Then (i) Velocity vector is φ(x, y) = ax by and ax by = c 1, are equipotentials. (ii) Stream line is ψ(x, y) = bx + ay and bx + ay = c 2, are the stream line flows. These lines are parallel and makes an angle θ = tan 1 (b/a). A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 15 / 16
Applications Potential Flow Example For the complex potential F(z) = B 3 z3, B > 0. Velocity potential is φ(x, y) = Bx 3 (x 2 3y 2 ) and the stream line is ψ(x, y) = By 3 (3x 2 y 2 ). A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 16 / 16