Question

Show that each of the following functions - call each one \(u(x, y)-\) is harmonic, and find the function's harmonic partner, \(v(x, y)\), such that \(u(x, y)+i v(x, y)\) is analytic. (a) \(x^{3}-3 x y^{2}\); (b) \(e^{x} \cos y\); (c) \(\frac{x}{x^{2}+y^{2}}, \quad\) where \(x^{2}+y^{2} \neq 0\); (d) \(e^{-2 y} \cos 2 x\); (e) \(e^{y^{2}-x^{2}} \cos 2 x y\); (f) \(e^{x}(x \cos y-y \sin y)+2 \sinh y \sin x+x^{3}-3 x y^{2}+y\).

Short answer

The harmonic partners \(v(x, y)\) of functions (a) \(x^{3}-3x y^{2}\), (b) \(e^{x} \cos y\), (c) \(\frac{x}{x^{2}+y^{2}}\), (d) \(e^{-2 y} \cos 2 x\), (e) \(e^{y^{2}-x^{2}} \cos 2 x y\) and (f) \(e^{x}(x \cos y-y \sin y)+2 \sinh y \sin x+x^{3}-3 x y^{2}+y\) are respectively (a) \(yx^{2} - 4y^{3}\), (b) \(e^{x} \sin y\), (c) \(\frac{y}{x^{2}+y^{2}}\), (d) \(e^{-2 y} \sin 2 x\), (e) \(e^{y^{2}-x^{2}} \sin 2 x y\) and (f) \(e^{x}(x \sin y+y \cos y)+2 \sinh y \cos x+3xy^{2}-x^{3}\).

Step-by-step solution

Test For Harmonicity

Confirm that each function \(u(x,y)\) is a harmonic function. This will involve taking second derivatives of the function with respect to x and y, and confirming that the sum of these second derivatives equals zero, which is in line with Laplace's equation \(\Delta u = \frac{\partial^2 u}{\partial x^2}+ \frac{\partial^2u}{\partial y^2}=0\). For any given function, say \(x^{3}-3x y^{2}\), we have \(\frac{\partial^2 u}{\partial x^2} = 6x\) and \(\frac{\partial^2 u}{\partial y^2} = -6y\). Adding these gives zero, hence the function is harmonic.

Finding The Harmonic Partner

The harmonic conjugate \(v(x, y)\) of a function \(u(x, y)\) results from integrating the Cauchy-Riemann equations, which are \(\frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}\) and \(\frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}\). If \(u(x, y)=x^{3}-3xy^{2}\) then \(\frac{\partial u}{\partial x}=3x^{2}-3y^{2}\) and \(\frac{\partial u}{\partial y}=-6xy\). Integrating the first equation with respect to y gives \(v(x, y) = yx^{2}-y^{3}+f(x)\) for some function f(x). Equating this to the integral of the second equation with respect to x gives \(f(x) = -3xy^{2}+g(y)\) for some function g(y). In this case, we take \(g(y) = 0\), hence the harmonic partner is \(v(x, y) = yx^{2} - y^{3} - 3xy^{2} = yx^{2}-4y^{3}\). This process is repeated for each of the given functions.

Key concepts

Laplace's equation

At the heart of harmonic functions lies Laplace's equation, a key staple in mathematical physics and engineering. When a function, such as a potential temperature distribution or electric potential, satisfies this equation, it is deemed to be 'harmonic.' The Laplace's equation is written as \[\Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0\]This neat equation is saying that, for a function to be harmonic, the sum of its second-order partial derivatives with respect to the spatial variables x and y should be zero. This points to a balance of sorts, indicating that the function is locally averaging out to be smooth and free from sudden changes in value, a characteristic of many physical phenomena. A function that meets this condition will exhibit symmetry and regularity, like the ripples on a calm pond, or the even hum of a well-tuned engine.

Harmonic conjugate

When exploring the realm of complex functions, there is a vital concept known as the harmonic conjugate. Given a harmonic function, say, \(u(x, y)\), there's a partner function \(v(x, y)\), that together form a complex function \(u + iv\) which is analytic. These pairs are like dance partners in the tango of complex analysis, each moving in step with the other through the Cauchy-Riemann equations. To find the harmonic conjugate of a function, we perform a dance of integration, guided by the Cauchy-Riemann equations, to ensure that every twist and turn of \(u\) is matched by \(v\). The search for \(v\) entails integrating and combining terms, where arbitrary functions may arise, only waiting to be determined by additional conditions or boundary values. It's almost like finding the other half of a destined pair, completing the picture to reveal the full, analytic function.

Cauchy-Riemann equations

The Cauchy-Riemann equations are the gatekeepers of the complex plane, ensuring that a function's passage into the realm of analyticity is by the books. These equations provide essential conditions for a function of a complex variable to be considered analytic. Written as a pair of first-order partial differential equations: \[\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \]and\[\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\]Here, \(u\) and \(v\) represent the real and imaginary parts, respectively, of an analytic function \(f(z) = u(x, y) + iv(x, y)\). These conditions ensure that the function is not only smooth but also conforms to the circular nature of complex multiplication. In essence, they are the mathematical handshake that seals the deal on differentiability in the complex plane.

Analytic functions

Analytic functions are the superstars of complex analysis. They're the kind of functions that are not only smooth and differentiable, but also able to be represented by a power series around every point in their domain. Think of them like the Swiss Army knives of the function world—utterly versatile and adaptable. For a function \(f(z)\) to be analytic, it must meet the Cauchy-Riemann equations and be differentiable throughout an open domain in the complex plane. The beauty of analytic functions lies not just in their differentiability, but also in their predictability; if you know an analytic function in a small region, you can predict it everywhere within its radius of convergence. This powerful property paves the way for techniques like contour integration, which unlocks a treasure trove of problem-solving tools in both mathematics and physics.