Problem 22 - Advanced Engineering Mathematics

Question

In Problems 13-22, expand the given function in a Taylor series centered at the indicated point. Give the radius of convergence of each series. $$ f(z)=e^{z}, z_{0}=3 i $$

Step-by-Step Solution

Answer

The Taylor series is \( \sum_{n=0}^{\infty} \frac{e^{3i}}{n!} (z-3i)^n \), and the radius of convergence is infinite.

Step 1: Recall the Taylor Series Formula

The Taylor series of a function \( f(z) \) centered at \( z_0 \) is given by: \[ f(z) = \sum_{n=0}^{fty} \frac{f^{(n)}(z_0)}{n!} (z-z_0)^n \] where \( f^{(n)}(z_0) \) is the \( n \)-th derivative of \( f(z) \) evaluated at \( z_0 \).

Step 2: Compute Derivatives of f(z)

For the function \( f(z) = e^z \), all derivatives are the same as the function itself:\[ f^{(n)}(z) = e^z \] Therefore, \( f^{(n)}(3i) = e^{3i} \) for all \( n \geq 0 \).

Step 3: Evaluate Derivatives at z_0

We need to evaluate the derivatives at the point \( z_0 = 3i \): For any \( n \), \( f^{(n)}(3i) = e^{3i} \).

Step 4: Write the Taylor Series Expression

Substituting into the Taylor series formula, we have: \[ f(z) = \sum_{n=0}^{\infty} \frac{e^{3i}}{n!} (z-3i)^n \]This is the Taylor series expansion of \( e^z \) centered at \( z_0 = 3i \).

Step 5: Find the Radius of Convergence

For the exponential function \( e^z \), the radius of convergence is infinite. Thus, the series converges for all \( z \).

Key Concepts

Radius of Convergence

The concept of the radius of convergence is crucial when dealing with series, such as Taylor or power series. In a power series, the radius of convergence tells us in which region around a center point that series is guaranteed to converge to the function it represents.

• When a series has an infinite radius of convergence, it converges for any value of the variable.
• For example, the exponential function, like the one in our problem, has an infinite radius of convergence.

For our specific example of the function \( f(z)=e^z \), the radius of convergence is infinite. This means the series will converge everywhere in the complex plane, no matter what \( z \) value you use.

This universal convergence is one of the remarkable properties of the exponential function and makes it a powerful tool in both real and complex analysis.

Derivative Computation

Understanding the computation of derivatives is at the heart of expanding functions into Taylor series. The derivative of a function essentially provides us with the slope of the function at any given point and is vital for constructing an accurate series representation.

• For the exponential function \( e^z \), its derivatives are quite straightforward.
• The key property is that all higher-order derivatives of \( e^z \) remain \( e^z \) itself.

This simplification due to the self-derivative feature makes it especially easy to work with when constructing Taylor or power series. Every derivative calculated at the point \( z_0 = 3i \) yields the same result: \( e^{3i} \).

Therefore, when we calculated the derivatives for the Taylor series expansion, the process is rendered straightforward. This characteristic allows the higher terms to be quickly deduced without additional computations.

Complex Exponential Function

The complex exponential function is an extension of the exponential function to complex numbers. It retains many of the properties of its real counterpart, with some additional important features.

• The function \( e^z \) can be expressed using Euler's formula as \( e^{z} = e^{x+iy} = e^{x} (\cos(y) + i\sin(y)) \).
• This expression connects the exponential function with trigonometric functions, expanding its utility to the field of complex numbers.

For instance, in our exercise with \( e^{3i} \), applying Euler's formula will give us:\[ e^{3i} = \cos(3) + i\sin(3) \]This highlights how exponential functions intertwine with angles and rotations in the complex plane, offering insights into phenomena like wave functions in physics or transformations in graphics.

Overall, the complex exponential standing as a staple in mathematical analysis shows its adaptability across various branches of mathematics and applied sciences.