Question

What is the largest circle within which the Maclaurin series for tanh \(z\) converges to \(\tanh z\) ?

Short answer

The largest circle within which the Maclaurin series for \(\tanh z\) converges to \(\tanh z\) is \(|z| < \frac{\pi}{2}\).

Step-by-step solution

Identify the singularities of \(\tanh z\)

\(\tanh z\) can be written as \(\frac{e^z - e^{-z}}{e^z + e^{-z}}\). This expression is undefined where the denominator is zero, which happens at \(z = \frac{\pi}{2} + k\pi i\), for \(k\) an integer.

Find the distance to the closest singularity

Since we are interested in the Maclaurin series around the origin, we need to find the distance from the origin to the closest singularity. This is \(\frac{\pi}{2}\).

Conclusion

The radius of convergence for the Maclaurin series of \(\tanh z\) is the distance to the closest singularity, which is \(\frac{\pi}{2}\). Therefore, the Maclaurin series converges to \(\tanh z\) within the circle \(|z| < \frac{\pi}{2}\).

Key concepts

Tanh z Function

The hyperbolic tangent function, denoted as \(\tanh z\), is a widely used function in mathematics, particularly in complex analysis. It is defined as \(\tanh z = \frac{\sinh z}{\cosh z}\) where \(\sinh z\) and \(\cosh z\) are the hyperbolic sine and cosine functions, respectively. Alternatively, using the exponential function, it can be expressed as \(\frac{e^z - e^{-z}}{e^z + e^{-z}}\).

This function exhibits properties similar to the familiar tangent function but operates on the hyperbolic plane. It is an odd function, meaning \(\tanh(-z) = -\tanh(z)\), and it asymptotically approaches \(1\) and \( -1\) as \(z\) approaches infinity and negative infinity, respectively, in the real part.

The \(\tanh z\) function plays an essential role in various fields, including physics and engineering, due to its behavior when modeling systems that rapidly reach a maximum or minimum value. Understanding its behavior in the complex plane, including its singularities and series representation, is fundamental for advanced studies in complex analysis and related areas.

Singularities in Complex Analysis

In complex analysis, singularities are points at which a function does not behave well, in the sense that it does not remain finite, differentiable, or even continuous. There are a few types of singularities, most notably poles, essential singularities, and removable singularities.

A pole occurs at a point in the complex plane where a function takes on infinite values. For the \(\tanh z\) function, poles occur where the denominator of the function \(\frac{e^z - e^{-z}}{e^z + e^{-z}}\) goes to zero. Specifically, the poles are located at \(z = \frac{\pi}{2} + k\pi i\), where \(k\) is an integer. At these points, the function \(\tanh z\) is not defined since it becomes infinitely large. This behavior deeply influences the convergence of series representations of the function in the complex plane.

Radius of Convergence

The radius of convergence is a measure of the extent to which a power series will converge, or indeed representative of its valid region. For a power series centered at a point \(a\) in the complex plane, the radius of convergence is the distance from \(a\) to the closest singularity of the function being represented.

In the case of the Maclaurin series for the \(\tanh z\) function, the series is centered at the origin \(z = 0\). Therefore, the radius of convergence will be the distance to the nearest singularity. Nonetheless, it should be emphasized that the Maclaurin series will faithfully represent the function \(\tanh z\) within the circle of convergence \( |z| < \frac{\pi}{2}\), based on the distance to the nearest pole. Outside this circle, the series may diverge or fail to accommodate the natural continuation of the function. By finding the radius, one can ensure the reliability of the series within its boundary.