Question

Let \(f\) be analytic within and on the circle \(\gamma_{0}\) given by \(\left|z-z_{0}\right|=r_{0}\) and integrated in the positive sense. Show that Cauchy's inequality holds: $$\left|f^{(n)}\left(z_{0}\right)\right| \leq \frac{n ! M}{r_{0}^{n}}$$ where \(M\) is the maximum value of \(|f(z)|\) on \(\gamma_{0}\).

Short answer

The proof of the Cauchy's inequality is accomplished by following the above steps, where we essentially leveraged Cauchy's integral formula and the triangle inequality.

Step-by-step solution

Recall Cauchy's integral formula

Cauchy's integral formula for the \(n\)-th derivative of \(f\) is \[f^{(n)}(z_{0}) = \frac{n!}{2 \pi i} \oint_{\gamma_{0}} \frac{f(z)}{(z-z_{0})^{n + 1}} dz\]

Take the magnitude on both sides

When taking magnitude on both sides, the right side becomes an inequality due to triangular inequality, hence \[|f^{(n)}(z_{0})| \leq \frac{n!}{2 \pi} \oint_{\gamma_{0}} \frac{|f(z)|}{|z-z_{0}|^{n + 1}} |dz|\]

Use the fact that \(|f(z)| \leq M\) and \(|z-z_{0}| = r_{0}\)

We obtain from the inequality the relation \(\frac{|f(z)|}{|z-z_{0}|^{n + 1}} \leq \frac{M}{r_{0}^{n+1}}\). Insert this into the inequality, we get \[|f^{(n)}(z_{0})| \leq \frac{n!M}{2 \pi r_{0}^{n}} \oint_{\gamma_{0}} |dz|\]

Simplify the right side

Notice that \(\oint_{\gamma_{0}} |dz| = 2\pi r_{0}\), using this we then simplify the right side of the inequality to obtain \[|f^{(n)}(z_{0})| \leq \frac{n! M}{r_{0}^{n}}\]

Key concepts

Analytic Functions

Analytic functions form the backbone of complex analysis, offering fascinating insights into mathematical behavior in complex domains. At their core, an analytic function is a complex function that is locally represented by a convergent power series. This means that around any point within its domain, you can express the function as a sum of powers of \(z\), which is often written as \(f(z) = a_0 + a_1(z - z_0) + a_2(z - z_0)^2 + \ldots\).

These functions are remarkable because they are infinitely differentiable within their domain and possess derivatives of all orders. The nature of analytic functions allows us to perform powerful operations, such as integrating along paths and applying series expansions. This smoothness and predictability enable various profound results and theorems like Cauchy's integral formula and Cauchy's inequality.

For a function to be analytical, it must satisfy certain conditions, such as being holomorphic, which implies it is differentiable at every point in its domain. Holomorphicity equates to the satisfaction of the Cauchy-Riemann equations, a cornerstone requirement for a function to be analytic in complex analysis. This differentiable property sets them apart from functions that might be merely continuous or piecewise continuous.

Cauchy's Integral Formula

Cauchy's integral formula is a pivotal result in complex analysis, offering a direct way to understand both the value and the derivatives of analytic functions. Simply put, it relates the value of a function inside a closed curve to a contour integral around that curve.

The formula is written as follows for the function \(f(z)\) at a point \(z_0\): \[f(z_0) = \frac{1}{2 \pi i} \oint_{\gamma} \frac{f(z)}{z-z_0} \, dz\] This expression tells us that the value of an analytic function at any point inside the curve \(\gamma\) can be determined by an integral over the function values along \(\gamma\).

An extended version of the formula gives us a way to find derivatives of any order: \[f^{(n)}(z_0) = \frac{n!}{2 \pi i} \oint_{\gamma} \frac{f(z)}{(z-z_0)^{n+1}} \ dz\]

This extended formula is particularly useful because it lays the foundation for proving more advanced results like Cauchy's inequality. By understanding how derivatives of different orders relate to contour integrals, many properties of analytic functions can be explored. It's this framework that allows complex analysis to marry geometry (via the contours) with algebra (via the function and its derivatives) in a profound and elegant symphony.

Complex Analysis

Complex analysis is a branch of mathematical analysis dedicated to the study of functions of a complex variable. This field explores complex numbers and their functions, providing deep insights due to the unique properties of complex numbers compared to real numbers.

One of the key features of complex analysis is its power to solve integrals and differential equations more straightforwardly. This stems from the fact that many concepts in complex analysis, like contour integration, aren't only theoretical tools but also practical ones used in physics and engineering.

Cauchy's theorem and its resultant formulas are central to this field, showing how integrals around closed loops can often simplify to zero for functions that are analytic within the contained domain. This can drastically reduce the complexity of solving problems.

• Complex analysis helps in evaluating real integrals using complex functions.
• It provides techniques like residue calculus, which can be used to compute integrals and solve complex differential equations.
• The field also enables a deeper understanding of convergence series, especially with complex power series.

Another remarkable result of complex analysis is the ability to express functions as power series through analytic continuation, an extension of power series beyond their radius of convergence. These abilities highlight why complex analysis is seen as a critical field in pure and applied mathematics alike.