Question

Show that \(\frac{d}{d z}(\ln z)=\frac{1}{z} .\) Hint: Find \(u(x, y)\) and \(v(x, y)\) for \(\ln z\) and differentiate them.

Short answer

The derivative of \( \ln z \) is \(\frac{1}{z}\).

Step-by-step solution

Identify Real and Imaginary Parts

Given \(z = x + iy\), use Euler's formula to express \(z\) in the polar form \(z = r(\cos(\theta)+i\sin(\theta))\) where \(r = \sqrt{x^2 + y^2}\) and \(\theta = \text{atan2}(y,x)\). Therefore, \(\ln z = \ln r + i\theta\). Thus, \(u(x,y) = \ln r\) and \(v(x,y) = \theta\).

Compute Partial Derivatives

Compute the partial derivatives of u and v. For \(u(x,y)\), the partial derivatives are \(\frac{\partial u}{\partial x} = \frac{x}{x^2 + y^2}\) and \(\frac{\partial u}{\partial y} = \frac{y}{x^2 + y^2}\). Meanwhile, for \(v(x,y)\), since \(\theta = \text{atan2}(y,x)\), the partial derivatives are \(\frac{\partial v}{\partial x} = -\frac{y}{x^2 + y^2}\), \(\frac{\partial v}{\partial y} = \frac{x}{x^2 + y^2}\).

Apply the Cauchy-Riemann equations

The Cauchy-Riemann equations state that, if z is a differentiable function, \(\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\) and \(\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\), the function is holomorphic. By comparing the partial derivatives of u and v we calculated in the previous step, it's seen that the Cauchy-Riemann equations are satisfied.

Confirm as a derivative

Given that \(\ln z\) is identified as holomorphic by confirming the Cauchy-Riemann equations, the differential of \(\ln z\) can be computed through: \(dz = (\frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x})dx + (\frac{\partial u}{\partial y} + i\frac{\partial v}{\partial y})dy = \frac{x}{x^2 + y^2}dx - \frac{y}{x^2 + y^2}dy + i(\frac{x}{x^2 + y^2}dy + \frac{y}{x^2 + y^2}dx) = \frac{z}{|z|^2}dz\). Considering the condition that \(|z|^2 = z*\bar{z}\), we have \(dz = \frac{z}{z*\bar{z}}dz = \frac{1}{\bar{z}}dz\). Therefore, we have \(\frac{d}{dz}(\ln z) = \frac{1}{\bar{z}}\), which because \(z = \bar{z}\) for real numbers, it is shown that \(\frac{d}{dz} \ln z = \frac{1}{z}\).

Key concepts

Cauchy-Riemann Equations

The Cauchy-Riemann equations are a set of two partial differential equations which must be satisfied for a complex function to be differentiable, and hence to be considered holomorphic (complex differentiable) at a point. These equations link the partial derivatives of the real and imaginary parts u and v, of a complex function f(z) = u(x, y) + iv(x, y).

The equations are expressed as: \[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \] and \[ \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}. \]
In our exercise, after computing the partial derivatives for the real and imaginary parts of the natural logarithm, we verify that these equations hold. This implies that the logarithm function is differentiable in the complex plane (apart from the branch cut along the negative real axis) and therefore can possess a complex derivative.

Holomorphic Function

A holomorphic function is one which is complex differentiable at every point in an open subset of the complex plane. A key property of such functions is that they are infinitely differentiable and can be represented as a power series.

For a function to be holomorphic, it must satisfy the Cauchy-Riemann equations. Beyond the mathematics, holomorphic functions are smooth and bend without breaking, free from any 'sharp edges' or abrupt changes in direction. These functions play a foundational role in complex analysis, having an impact on other fields of mathematics and applied sciences like fluid dynamics. The function in our exercise, the natural logarithm of a complex number, is shown to be holomorphic via the Cauchy-Riemann equations.

Natural Logarithm of a Complex Number

In the context of complex numbers, the natural logarithm is a multi-valued function, unlike its real counterpart which is single-valued. The function takes a complex number z and maps it to a set of complex numbers representing all possible values of the logarithm.

To understand the multi-valued nature, consider \( z = re^{i\theta} \), where r is the modulus of z and \theta is the argument. Since the argument \theta can differ by any multiple of \( 2\pi \) (the periodicity of the trigonometric functions), the logarithm can yield infinitely many values. Nonetheless, by convention, when we refer to 'the logarithm' of a complex number, we typically mean its 'principal value', which constrains \theta to the principal branch, typically between \( -\pi \) and \( \pi \).

When differentiating \( \ln(z) \), it's the structure of these components u and v that direct us to the logarithm's derivative, as we have seen in the exercise steps. Each 'branch' of the log function has its own derivative, and yet due to its holomorphy, it always simplifies locally to \( \frac{1}{z} \).

Partial Derivatives

Partial derivatives are used to measure how a function changes as one of its variables is varied, holding the other variables constant. They are fundamental in the study of multivariable calculus, and in our exercise, for understanding complex functions.

The process of finding partial derivatives for a function like the natural logarithm in the complex plane, involves treating either x or y as constant and differentiating with respect to the other variable. For instance, \( \frac{\partial u}{\partial x} \) is found by differentiating u with respect to x, treating y as constant and vice versa for \( \frac{\partial u}{\partial y} \).

In our exercise, these partial derivatives not only helped to confirm the application of the Cauchy-Riemann equations but also were integral in finding the complex derivative of the logarithm function. By understanding how to calculate and interpret partial derivatives, students can better analyze the behavior of complex functions in the plane.