Complex Analysis: Computing Line Integrals Step-by-Step

Parameterization expresses curve coordinates as functions of a single parameter t. For line segments between P₀ and P₁: r(t) = P₀ + t(P₁ - P₀), t ∈ [0,1]. For circles: x(t) = r cos(t), y(t) = r sin(t), t ∈ [0, 2π]. For spirals: x(t) = r(t) cos(t), y(t) = r(t) sin(t), where r(t) varies with t. The parameter range determines how many times the curve is traversed.

To parametrize a line segment from point P₀(x₀, y₀) to point P₁(x₁, y₁): use P₀ as the starting point, use the vector from P₀ to P₁ as the direction vector: (x₁ - x₀, y₁ - y₀), with the parametrization x = x₀ + (x₁ - x₀)t, y = y₀ + (y₁ - y₀)t, with t restricted to [0, 1]. Circles cannot be functions (fail vertical line test). Standard parametrization: x = r·cos(t), y = r·sin(t). Ellipses: x = a·cos(t), y = b·sin(t). The same expression appears in both trig functions. Direction: cosine for x and sine for y gives counterclockwise traversal; sine for x and cosine for y gives clockwise.

This section covers how to represent curves using parametric equations. A curve can be expressed as x = x(t) and y = y(t), or as a vector function r(t) = <x(t), y(t)>. The parameter t imposes an orientation on the curve—the direction in which t increases. Examples include: (1) Circles of radius 5: counterclockwise uses x = 5cos(t), y = 5sin(t); clockwise uses x = 5cos(t), y = -5sin(t). (2) Functions y = f(x): let x = t, y = f(t). (3) Line segments from P₀(x₀,y₀) to P₁(x₁,y₁): use direction vector v = <x₁-x₀, y₁-y₀>, set x = x₀ + v_x*t, y = y₀ + v_y*t, and restrict t to [0,1].

A parametrization describes a curve by expressing the x and y coordinates as functions of a parameter t, where the coordinate functions x(t) and y(t) give the position of a point on the curve at each parameter value, and the parameter interval ensures the curve is traced exactly once without overlapping (except possibly at endpoints). For example, a circle with radius 2 and center (4,3) can be parametrized by x(t) = 4 + 2cos(t), y(t) = 3 + 2sin(t) for t in (0, 2π), while a line segment from (2,4) to (5,-1) can be parametrized by x(t) = 2 + 3t, y(t) = 4 - 5t for t in (0,1).

This extensive section provides systematic instruction on constructing fundamental geometric shapes using parametric equations. It covers three essential curve types: line segments between two points, circles, and ellipses. For line segments, the formula x(t) = xmin + (xmax - xmin)*((t - tmin)/(tmax - tmin)) enables interpolation between endpoints. Circles are derived from the unit circle parametrization with center shifting: x(t) = x0 + r*cos(t), y(t) = y0 + r*sin(t). Ellipses extend this concept with separate x and y radii: x(t) = x0 + rx*cos(2π((t - tmin)/(tmax - tmin))), y(t) = y0 + ry*sin(2π((t - tmin)/(tmax - tmin))). The section emphasizes that circles are special cases where rx = ry. These building blocks form the essential vocabulary for more complex curve constructions in subsequent sections.

This lecture introduces vector fields and scalar line integrals as steps toward understanding the generalization of the fundamental theorem of calculus. The fundamental theorem of calculus connects differentiation and integration, and in multivariable calculus, this connection is extended through Stokes' formula, which states that for a compact manifold with boundary M and differential form ω, the integral of ω on the boundary equals the integral of dω on the interior. Green's theorem is a special case: for a planar domain D with boundary ∂D, the line integral of Pdx + Qdy over ∂D equals the double integral of (Q_x - P_y) over D. A vector field assigns a vector to each point in its domain, represented as F(x,y) = P(x,y)i + Q(x,y)j in 2D. A vector field is conservative if it equals the gradient of a scalar function f, called the potential. For F = (P,Q) to be conservative, the condition P_y = Q_x must hold (Clairaut's theorem). A scalar line integral ∫_C f ds is computed by parameterizing the path as r(t) for t in [a,b], then ∫_a^b f(r(t)) |r'(t)| dt. The integral of 1 along C gives the length of C, so dividing by length gives the average value of f on C.

A line integral for real-valued functions computes the integral of a vector field along a curve gamma. The curve maps from a real interval to R², and the integral is computed by composing the function with gamma and multiplying by the derivative of gamma. For the integral to exist, gamma must be C¹ (derivatives exist and are continuous) and the vector field must be continuous. Curves are often parameterized using a parameter t, such as an ellipse given by x = a cos(t), y = b sin(t) for t in [0, 2π]. The computation involves substituting the parametric equations into the vector field and integrating with respect to t.

Line integrals over vector fields extend single-variable calculus to two dimensions. A vector field F = (P, Q) acts on a curve C parameterized by R(t) = (x(t), y(t)). The integral ∫C F · dR converts to ∫A^B [P(x(t), y(t)) * dx/dt + Q(x(t), y(t)) * dy/dt] dt. Alternatively, this is written as ∫C P dx + Q dy. For circles, use polar coordinates: R(t) = (r cos t, r sin t) with t ∈ [0, 2π]. Single points where parameterization repeats do not affect integral values as they have measure zero.

Line integrals of scalar functions f(x,y) over a curve C are computed as ∫_C f(x(t),y(t))√[(dx/dt)²+(dy/dt)²]dt, representing the accumulation of f values along the curve weighted by arc length. For vector fields F = P i + Q j, the line integral ∫_C F·dr = ∫_C (P dx + Q dy) measures work done by the field along the path. The Fundamental Theorem of Line Integrals states that if F = ∇f is a conservative vector field, then ∫_C ∇f·dr = f(r(b)) - f(r(a)), making the integral independent of path and zero for closed curves. A vector field is conservative if and only if ∂P/∂y = ∂Q/∂x on a simply connected region.

Line integrals are divided into scalar field and vector field types. For scalar fields, curves in R2 or R3 can be parameterized using t from a to b, with position vector r(t) = x(t)i + y(t)j. The line integral of f(x,y) along C is defined using a Riemann sum: divide [a,b] into subintervals, compute f(x(t_k*), y(t_k*)) times arc length Δs_k, sum, and take limit as n→∞. To calculate, parameterize the curve, substitute into f, and express ds = √[(dx/dt)² + (dy/dt)²] dt. When f(x,y) > 0, the integral represents area between the curve and x-axis. When the curve is the x-axis, it reduces to the standard single-variable integral, showing line integrals generalize single integrals to curved paths.

A complex function f(z) is holomorphic at a point if it is complex differentiable there, meaning it can be approximated by a linear function f(z) ≈ f(z₀) + a(z - z₀) + error, where the error divided by |z - z₀| tends to zero. For a function w = u + iv of z = x + iy to be holomorphic, its real and imaginary parts must satisfy the Cauchy-Riemann equations: ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x. These equations ensure that the Jacobian matrix of partial derivatives corresponds to multiplication by a complex number a = ∂u/∂x + i∂v/∂x. Holomorphic functions are also called analytic functions in complex analysis, and they possess remarkable properties such as being infinitely differentiable and having convergent power series expansions.

A complex function f(z) = u(x,y) + iv(x,y) is complex differentiable (holomorphic) at a point if and only if its partial derivatives exist, are continuous, and satisfy the Cauchy-Riemann equations (∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x) at that point; such functions are called analytic, and if they are analytic everywhere in the complex plane, they are called entire functions.

A function is holomorphic (analytic) at a point if it is differentiable in a neighborhood of that point. Holomorphic functions are also called analytic functions. The key properties of holomorphic functions include: (1) They are infinitely differentiable, (2) They can be represented by power series in a neighborhood of each point, (3) They satisfy the Cauchy-Riemann equations, (4) They are continuous. The distinction between holomorphic and merely differentiable functions is important: a function can be differentiable at a point without being holomorphic (analytic) in a neighborhood. Holomorphic functions are the central objects of study in complex analysis.

A complex function f(z) = u(x,y) + iv(x,y) is analytic if it satisfies the Cauchy-Riemann equations: ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x. For example, f(z) = z̄ (complex conjugate) is not analytic because ∂u/∂x = 1 ≠ ∂v/∂y = -1, while f(z) = e^z, f(z) = z², and f(z) = cos(e^iz) are all analytic as they satisfy both equations.

An analytic function is a complex function f(z) = u + iv that is differentiable at a point z = a, and it must satisfy the Cauchy-Riemann equations: ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x. If a function is not differentiable at a point, it is called a singular function, and that point is a singularity. An entire function is analytic throughout the entire z-plane. The Milne-Thompson method is used to form an analytic function when given either the real or imaginary part by applying the Cauchy-Riemann equations and integrating.