Let \(u(x,y)\) be harmonic and have continuous partial derivatives of order three in an open disc of radius \(R>0\).
Let two points \((a,b), (x,y)\) in this disk be given. Show that the following integral is independent of the path in this disk joining these points: \[ v(x,y) = \int_{a,b}^{x,y} ( -\frac{\partial u}{\partial y}dx + \frac{\partial u}{\partial x}dy) .\]
In parts:
- Prove that \(u(x,y)+i v(x,y)\) is an analytic function in this disc.
- Prove that \(v(x,y)\) is harmonic in this disc.
Solution
Goal: Let \(D = D((x_0, y_0), R) \subset \mathbb{R}^2\) be an open disk, and let \(u \in C^3(D)\) be a real-valued harmonic function (\(\Delta u = u_{xx} + u_{yy} = 0\)). 1. (a) Prove that the differential 1-form \(\omega = -u_y \, dx + u_x \, dy\) is closed, and that the line integral \(\int_{(a,b)}^{(x,y)} \omega\) is independent of the path in \(D\). 2. (b)(i) Prove that \(f(z) = u(x,y) + i v(x,y)\) is analytic (holomorphic) in \(D\). 3. (b)(ii) Prove that \(v(x,y)\) is harmonic in \(D\).
Part (a): Path Independence of the Line Integral
1. The differential form \(\omega = P\,dx + Q\,dy\) with \(P = -u_y\) and \(Q = u_x\) is \(C^1\) and closed on \(D\).
1.1. Since \(u \in C^3(D)\), the partial derivatives \(P = -u_y\) and \(Q = u_x\) are in \(C^2(D)\), hence \(C^1(D)\).Proof: Continuous differentiability of orders up to 3.
1.2. Compute the curl / exterior derivative:\[\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = \frac{\partial}{\partial x}(u_x) - \frac{\partial}{\partial y}(-u_y) = u_{xx} + u_{yy} = \Delta u.\]Proof: Clairaut’s theorem on equality of mixed partials and direct differentiation.
1.3. Since \(u\) is harmonic on \(D\), \(\Delta u = 0\), which implies \(\frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y}\) everywhere in \(D\).Proof: Hypothesis that \(u\) is harmonic.
1.4. Q.E.D.
2. Closed 1-forms on simply connected domains are exact, hence path independent.
2.1. An open disk \(D\) in \(\mathbb{R}^2\) is convex, hence simply connected.Proof: Standard geometric property of open disks.
2.2. By Green’s Theorem (or Poincaré’s Lemma), the line integral of a \(C^1\) closed 1-form along any closed piecewise smooth curve \(\gamma\) contained in \(D\) is zero: \(\oint_\gamma \omega = 0\).Proof: \(\oint_\gamma P\,dx+Q\,dy = \iint_{\text{Int}(\gamma)} (Q_x - P_y)\,dA = \iint_{\text{Int}(\gamma)} 0\,dA = 0\).
2.3. If \(\gamma_1, \gamma_2\) are any two piecewise smooth paths in \(D\) from \((a,b)\) to \((x,y)\), then \(\gamma_1 - \gamma_2\) is a closed loop, so \(\int_{\gamma_1} \omega - \int_{\gamma_2} \omega = \oint_{\gamma_1 - \gamma_2} \omega = 0 \implies \int_{\gamma_1} \omega = \int_{\gamma_2} \omega\).Proof: Additivity of path integrals.
2.4. Therefore, the function \(v(x,y) = \int_{(a,b)}^{(x,y)} (-u_y\,dx + u_x\,dy)\) is well-defined and independent of path in \(D\).Proof: Follows from
2.3.
2.5. Q.E.D.
Part (b)(i): Holomorphicity of \(u(x,y) + i v(x,y)\)
3. Compute the partial derivatives of \(v(x,y)\).
3.1. By the Fundamental Theorem of Calculus for line integrals of exact forms, \(\nabla v(x,y) = (P(x,y), Q(x,y)) = (-u_y(x,y), u_x(x,y))\).Proof: Differentiating \(v(x,y) = \int_{(a,b)}^{(x,y)} (P\,dx + Q\,dy)\) with respect to \(x\) and \(y\) along axis-parallel segments.
3.2. Explicitly, \(v_x = -u_y\) and \(v_y = u_x\).Proof: Extracting components from
3.1.
3.3. Q.E.D.
4. \(f(z) = u(x,y) + i v(x,y)\) satisfies the Cauchy-Riemann equations in \(D\).
4.1. The Cauchy-Riemann equations are \(u_x = v_y\) and \(u_y = -v_x\).Proof: Standard definition.
4.2. From <1>3.<2>2, \(v_y = u_x\) and \(v_x = -u_y \iff u_y = -v_x\).Proof: Direct comparison with <1>3.<2>2.
4.3. Since \(u \in C^3(D)\), its partial derivatives \(u_x, u_y, v_x, v_y\) are in \(C^1(D)\) and are continuous.Proof: Real differentiability follows from continuity of first partial derivatives.
4.4. A complex function \(f = u+iv\) whose real and imaginary parts are \(C^1\) and satisfy the Cauchy-Riemann equations is analytic (holomorphic) on \(D\).Proof: Standard characterization of complex differentiability.
4.5. Q.E.D.
Part (b)(ii): Harmonicity of \(v(x,y)\)
5. \(v(x,y)\) satisfies the Laplace equation \(\Delta v = 0\) on \(D\).
5.1. Since \(v \in C^2(D)\), compute its second partial derivatives using <1>3.<2>2:\[v_{xx} = \frac{\partial}{\partial x}(v_x) = \frac{\partial}{\partial x}(-u_y) = -u_{yx}.\]\[v_{yy} = \frac{\partial}{\partial y}(v_y) = \frac{\partial}{\partial y}(u_x) = u_{xy}.\]Proof: Differentiating <1>3.<2>2 with respect to \(x\) and \(y\).
5.2. Since \(u \in C^3(D) \subseteq C^2(D)\), Clairaut’s theorem guarantees \(u_{yx} = u_{xy}\).Proof: Equality of mixed partials for \(C^2\) functions.
5.3. Therefore, \(\Delta v = v_{xx} + v_{yy} = -u_{yx} + u_{xy} = 0\).Proof: Substitution of 2 into
5.1.
5.4. Thus, \(v(x,y)\) is harmonic in \(D\).Proof: Definition of harmonic function.
5.5. Q.E.D.