Question

Show that \(d \Gamma(z+1) / d z\) exists and is finite by establishing the following: (a) \(|\ln t|0\). Hint: For \(t \geq 1\), show that \(t-\ln t\) is a monotonically increasing function. For \(t<1\), make the substitution \(t=1 / s\). (b) Use the result from part (a) in the integral for \(d \Gamma(z+1) / d z\) to show that \(|d \Gamma(z+1) / d z|\) is finite. Hint: Differentiate inside the integral.

Short answer

The inequality \(|\ln t|0\) can be proved using the concept of monotonic functions and substitution which in turn helps to show that the function \(|d \Gamma(z+1) / d z|\) is finite.

Step-by-step solution

Prove \(|\ln t|0\)

The inequality can be solved in two parts. For \(t \geq 1\), we will make use of the hint and show that \(t-\ln t\) is a monotonically increasing function. This can be proved by taking the derivative of \(t-\ln t\), which is \(1-1/t\). Since the derivative is always positive for \(t \geq 1\), we can say that \(t-\ln t\) is monotonically increasing, which implies \(|\ln t|<t\). The other part is for \(t<1\). We can substitute \(t=1/s\) and repeat the process above to prove \(|\ln t|<1+1/t\).

Differentiate inside the integral

The Gamma function, \(\Gamma(z+1)\), is defined as \(\Gamma(z+1)=\int_{0}^{\infty} t^{z} e^{-t} d t\). By differentiating both sides with respect to \(z\) we get, \(d \Gamma(z+1) / d z=\int_{0}^{\infty} \frac{d}{d z}\left[t^{z} e^{-t}\right] d t\). Using product rule the differentiating inside the integral results in \(d \Gamma(z+1) / d z=\int_{0}^{\infty} \left(e^{-t} t^{z}\right)^{\prime} dt = \int_{0}^{\infty}(t^{z} e^{-t} \ln t-t^{z} e^{-t}) dt\).

Show \(|d \Gamma(z+1) / d z|\) is finite

From the absolute of the integral of the derivative, we can use the previously proven inequality to show the result is finite. Therefore, using \(|\ln t|<t+1 / t\), we find that \(|d \Gamma(z+1) / d z| < \int_{0}^{\infty} (t^{z+1}e^{-t}+t^{z-1}e^{-t}) dt\). And these are the integrals of \( \Gamma(z+2) \) and \( \Gamma(z) \) that are finite. Hence \(|d \Gamma(z+1) / d z|\) is also finite.

Key concepts

Gamma Function Properties

The Gamma function, denoted as \(\Gamma(z)\), is an extension of the factorial function to complex numbers, with its argument \(z\) shifted down by 1. That is, for a positive integer \(n\), \(\Gamma(n) = (n-1)!\).

One of the remarkable properties of the Gamma function is its recursive nature, where for any complex number \(z\) that is not a non-positive integer, it holds that \(\Gamma(z+1) = z\Gamma(z)\). This mirrors the factorial property \(n! = n \times (n-1)!\).

Another significant property is the Gamma function's differentiability. As shown in the exercise, by applying integration by parts, we differentiate through an integral and establish the derivative's existence. This differentiability also leads to the function being infinitely differentiable or smooth.

Moreover, the Gamma function is analytic everywhere, except at non-positive integers where it has simple poles. This means the function can be locally expressed as a power series around any point in its domain, which is a key characteristic in complex analysis. The behavior around its poles and how residues are calculated become a topic of interest in higher mathematics.

As a final point of this section, it must be stated that the Gamma function is not elementary, which means it cannot be expressed in terms of basic algebraic, exponential, or logarithmic functions alone. Despite this, it plays a crucial role in various areas of mathematics, physics, and engineering due to its utility in dealing with factorials of non-integer values and its occurrence in various integral forms.

Integration in Complex Analysis

Integration in complex analysis, notably line integrals and contour integrals, is much more nuanced than in real analysis. The key difference arises from the complex plane's structure, where contours can enclose singular points and switch between different branches of multi-valued functions.

The exercise touches upon the integral form of the Gamma function, which is an improper integral over the positive real line. When dealing with complex-valued functions, one must consider how the domain of integration, or contour, is affected by the singularity of functions.

In this context, the exercise demonstrates the differentiation under the integral sign, which is a powerful technique derived from Leibniz's integral rule. This method allows one to take the derivative of an integral whose limits are constant but the integrand includes a parameter. Specifically, it aids in confirming the existence of the derivative of a function defined by an integral, such as the Gamma function.

The complexities of integration in the complex plane also lead to profound results like Cauchy's integral theorem and Cauchy's integral formula. These results hinge on the analyticity of functions, similar to how the Gamma function was shown to be differentiable due to its properties under integration.

Monotonic Functions

Monotonic functions are functions that preserve a consistent order. A function is considered monotonically increasing if, for any two elements \(x_1 < x_2\), the function value at \(x_1\) is less than or equal to the function value at \(x_2\), denoted by \(f(x_1) \leq f(x_2)\). Similarly, a function is monotonically decreasing if the reverse inequality holds.

In the provided exercise, the function under consideration is \(t - \ln(t)\) for \(t \geq 1\) and its behavior indicates it is monotonically increasing, proven by showing its derivative is positive. These types of functions are beneficial in mathematics since their limits at infinity and continuity lead to important implications.

The concept of a monotonic function is valuable in establishing inequalities, which are critical in proving the boundedness of other functions or integrals. The relation derived using the monotonicity of \(t - \ln(t)\) is essential in proving that \(d \Gamma(z+1) / d z\) is finite, illustrating the link between monotonicity and the properties of certain special functions in mathematical analysis.