Problem 15 - Complex Analysis for Mathematics and Engineering

Question

Compute the Taylor series for the principal logarithm \(f(z)=\log z\) expanded about the center \(z_{0}=-1+i\).

Step-by-Step Solution

Answer

The Taylor series is \(\log(-1+i) + \frac{-1-i}{2}(z + 1 - i) + \frac{1}{4}(z + 1 - i)^2 + \ldots\)

Step 1: Identify the Function and Center

We're asked to find the Taylor series for the complex function \(f(z) = \log z\) centered at \(z_{0} = -1 + i\). A Taylor series expansion of a function \(f(z)\) about a center \(z_0\) is given by \(\sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z - z_0)^n\).

Step 2: Calculate the Principal Logarithm at Center

The principal logarithm of a complex number \(z = x + yi\) is given by \(\log z = \log |z| + i\arg z\), where \(\arg z\) is the argument in the range \((-\pi, \pi]\). For \(z_0 = -1 + i\), \(|z_0| = \sqrt{2}\) and \(\arg z_0 = \frac{3\pi}{4}\). Thus, \(\log(-1+i) = \log \sqrt{2} + i\frac{3\pi}{4}\).

Step 3: Find the Derivatives of \(f(z)\)

The first derivative of \(f(z) = \log z\) is \(f'(z) = \frac{1}{z}\). Higher-order derivatives are derivatives of \(\frac{1}{z}\): \(f''(z) = -\frac{1}{z^2}\), \(f'''(z) = \frac{2}{z^3}\), etc. The \(n\)-th derivative is \((-1)^{n-1}(n-1)!/z^n\).

Step 4: Evaluate the Derivatives at \(z_0\)

Evaluate the derivatives at \(z_0 = -1 + i\):- \(f'(z_0) = \frac{1}{-1+i} = \frac{-1-i}{2}\),- \(f''(z_0) = -\frac{1}{(-1+i)^2} = \frac{i}{2}\),- \(f'''(z_0) = \frac{2}{(-1+i)^3} = \frac{-1+3i}{4}\), etc.

Step 5: Construct the Taylor Series

Substitute the derivatives evaluated at \(z_0\) into the Taylor series formula:\[f(z) = \log(-1+i) + \sum_{n=1}^{\infty} \frac{(-1)^{n-1}(n-1)!}{n!} \left(\frac{1}{(-1+i)^n}\right) (z - (-1+i))^n\]Simplifying, using partial calculations for each coefficient, gives the expansion.

Step 6: Final Expression

The Taylor series can now be expressed as:\[\log(-1+i) + \left(\frac{-1-i}{2}\right)(z + 1 - i) + \frac{1}{4}(z + 1 - i)^2 + \ldots\]

Key Concepts

Complex Logarithm

The complex logarithm function is a fascinating extension of its real counterpart into the realm of complex numbers. When dealing with complex numbers, the logarithm is not as straightforward as its real version due to the multi-valued nature of the complex exponential function. For a complex number written in polar form, \(z = re^{i\theta}\), the principal logarithm is defined as \(\log z = \log |z| + i\arg z\). This expression consists of two parts:

• The magnitude of \(z\), represented by \(\log |z|\), which determines the distance from the origin in the complex plane.
• The angle or argument \(\arg z\), which specifies the direction from the positive real axis, confined to the interval \((-\pi, \pi]\).

When calculating the principal logarithm for a specific point, like \(z_0 = -1 + i\), we first find the modulus \(|z_0| = \sqrt{2}\) and the argument \(\arg z_0 = \frac{3\pi}{4}\). Substituting these into the principle logarithm formula gives \(\log(-1+i) = \log \sqrt{2} + i\frac{3\pi}{4}\). This calculation sets the stage for deriving the Taylor series of the complex function.

Complex Analysis

Complex analysis is a branch of mathematics exploring functions of complex numbers. Differentiating functions in this domain involves understanding how the function behaves in the complex plane—a two-dimensional space incorporating both real and imaginary parts of complex numbers.
The Taylor series is a key tool in complex analysis. It provides a way to express complex functions as an infinite sum of terms calculated from the function's derivatives at a specific point, known as the center of the series. For example, if we expand \(f(z) = \log z\) about \(z_0 = -1+i\), we construct the series:

• Begin with \(f(z_0)\), the logarithm at the center.
• Add terms comprising successive derivatives of \(f(z)\), evaluated at \(z_0\), multiplied by corresponding powers of \((z - z_0)\).

Such expansions are vital for simplifying complex functions, aiding in calculations, and providing insights into the function's behavior around a specified point.

Derivatives of Complex Functions

Derivatives in complex analysis extend the concept of differentiation from real numbers to complex functions. These derivatives provide critical insight into how complex functions change as their input varies. For a complex function \(f(z) = \log z\), the derivatives are derived as follows:

• The first derivative, \(f'(z) = \frac{1}{z}\), describes the rate of change of the logarithmic function at any point \(z\).
• Higher derivatives are systematic expansions based on the differentiation of \(f'(z)\). The second derivative, for example, becomes \(f''(z) = -\frac{1}{z^2}\), and follows a calculated pattern involving increments of exponent reversal for each higher order.
• The general \(n\)-th derivative formula, \((-1)^{n-1}(n-1)!/z^n\), determines all higher derivatives, providing an efficient means to calculate and construct the Taylor series.

Evaluating these derivatives at points like \(z_0 = -1 + i\) allows the formation of a detailed and accurate representation of \(f(z)\) in its Taylor series form, furnishing deep understanding in calculating complex logarithmic functions.