The function arctan(z) is the inverse of the tangent function, defined for complex numbers. It takes a complex number z and returns a complex angle whose tangent is z. This function is multivalued due to the periodic nature of the tangent function, leading to the necessity of branch cuts in its representation.
congrats on reading the definition of arctan(z). now let's actually learn it.
The function arctan(z) can be expressed using logarithms: $$arctan(z) = \frac{1}{2i} \ln\left(\frac{1+iz}{1-iz}\right)$$, which highlights its multivalued nature.
The principal value of arctan(z) is often taken with a branch cut along the imaginary axis from -i to i.
As z approaches infinity, arctan(z) approaches $$\frac{\pi}{2}$$, showing that it has a limit as one of its properties.
The derivative of arctan(z) is given by $$\frac{1}{1+z^2}$$, which indicates the rate of change of the function.
The values of arctan(z) are not unique; for any integer k, $$arctan(z) + k\pi$$ is also a valid value due to the periodic nature of the tangent function.
Review Questions
How does the definition of arctan(z) as an inverse function relate to its multivalued nature?
Arctan(z) being the inverse of the tangent function inherently leads to multivalued outputs because tangent is periodic. For any input z, the tangent function can yield the same value at multiple angles separated by integer multiples of $$\pi$$. Thus, when defining arctan(z), we have to consider all these angles, making it multivalued unless we impose restrictions like branch cuts.
Discuss how branch cuts affect the calculation of arctan(z) in complex analysis.
Branch cuts are essential when dealing with arctan(z) as they limit the range of values that the function can take, allowing us to select a principal value. Without these cuts, we could end up with multiple values for the same input. The standard approach places a branch cut along the imaginary axis from -i to i, ensuring that the values remain consistent and manageable while calculating arctan(z).
Evaluate the implications of choosing different branch cuts on the values returned by arctan(z) and how this affects complex analysis.
Choosing different branch cuts for arctan(z) can lead to different principal values for the same complex number, which ultimately affects continuity and differentiability in complex analysis. For instance, if we were to position our branch cut differently than along the imaginary axis, we would change how we calculate limits and derivatives around that cut. This choice can also impact applications involving analytic continuation and integration along paths that cross these cuts, creating complications in how we interpret results in complex settings.
The specific single-valued selection of a multivalued function, often used to define functions like arctan(z) in a standard way.
Multivalued Function: A function that can produce multiple outputs for a single input, commonly seen in inverse trigonometric and logarithmic functions.