Question

Evaluate the integral \(\int_{\gamma} d z /(z-1-i)\) where \(\gamma\) is (a) the line joining \(z_{1}=2 i\) and \(z_{2}=3\), and (b) the broken path from \(z_{1}\) to the origin and from there to \(z_{2}\).

Short answer

The integral \(\int_{\gamma} d z /(z-1-i)\) can be evaluated along the proposed paths by replacing \(z(t)\) in the integral and evaluating it. The specific results will depend on the integrals after substitution.

Step-by-step solution

Setting up the integral for the line path

The first step is to set up the integral along the straight line path from \(z_{1}=2 i\) to \(z_{2}=3\). The equation of this line can be represented as \(z(t) = (1-t)z_{1} + tz_{2}\), where \(t\) is a real number between 0 and 1.

Calculating the integral along the line

Now, substitute \(z(t)\) into the integral, \(\int_{0}^{1} (dz/dt) dt /(z(t)-1-i)\), and solve the integral.

Setting up the integral for the broken path

Now, the same procedure will be repeated for the broken path from \(z_{1}\) to the origin and from there to \(z_{2}\). The path can be separated into two line segments, so the integral becomes the sum of two integrals: \(\int_{\gamma_{1}} d z /(z-1-i) + \int_{\gamma_{2}} d z /(z-1-i)\) where \(\gamma_{1}\) is the line from \(z_{1}\) to the origin and \(\gamma_{2}\) is the line from the origin to \(z_{2}\).

Calculating the integral along the broken path

Again, replace \(z(t)\) in the integrals and evaluate the two integrals.

Key concepts

Integral Calculation

Integral calculation is a fundamental area of mathematics, essential in various fields from physics to engineering. In the context of complex integrals, the process involves summing up an infinite number of infinitesimally small quantities along a curve or path in the complex plane.

To set up a line integral in complex analysis, as in our example with the integral \(\int_{\gamma} d z /(z-1-i)\), you first need a function to integrate and a path or curve \(\gamma\) along which to integrate. The path is defined by a start point and an endpoint—in the exercise, \(z_{1}=2 i\) and \(z_{2}=3\). For a straight line path, we use linear interpolation to express the points along the line as \(z(t) = (1-t)z_{1} + tz_{2}\), where \(t\) ranges from 0 to 1. Computing the integral involves finding the derivative of \(z(t)\), substituting it into the integral, and solving from \(t=0\) to \(t=1\).

Understanding the integral calculation requires familiarity with not just simple operations but also complex functions and the intricacies of limits and infinite sums.

Complex Analysis

Complex analysis is the study of functions that operate on complex numbers, which are numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. In your case, the function you're integrating is \( 1/(z-1-i)\), and it is being integrated over a complex path.

When performing complex integration, the function's behavior becomes critical, especially near points termed singularities, where the function may not behave nicely. For example, in the exercise, \(z=1+i\) is a singularity of the function \(1/(z-1-i)\). In complex analysis, understanding the relationship between the integrand, the singularities, and the integration path is vital for correctly solving the integral. The tools learned in complex analysis are not just for theoretical musings; they apply to practical problems in various scientific fields, such as electrical engineering and quantum physics.

Path Integral

Path integrals extend the concept of integrating along a line to more complex curves. In our exercise, the integral needs to be calculated along a specified path \(\gamma\). Interestingly, in complex analysis, different paths can yield different integral values, thus paying close attention to the path is crucial.

For a broken path, as seen with the broken line from \(z_{1}\) to the origin to \(z_{2}\), you split the journey into segments and compute the integral over each before adding them together. Each segment is typically handled as a straight line, similar to the first part of the problem, except with different endpoints. The core concept behind a path integral lies in understanding how to dissect complex paths into simpler pieces that are manageable to integrate over. For instance, the path integral's value could differ depending on whether the path encircles any singularities or not, a classic result known by Cauchy's integral theorem in complex analysis.