Question

Use the series definition (2.72) of $e^z$ to prove that ${}^{12}d{e}^z/dz=e^z$.

Short answer

The derivative of $e^z$ is $e^z$.

Step-by-step solution

Recall the series definition of exponential function

The exponential function for a complex number $z$ is defined by the series:$$e^z=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{n!}.$$ This series represents the exponential function $e^z$ in terms of the series expansion.

Differentiate the series term by term

To differentiate $e^z$ with respect to $z$, we differentiate each term of the series:$$\frac{d}{dz}\left(\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{n!}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{d}{dz}\left(\frac{z^n}{n!}\right).$$The derivative of each term $\frac{z^n}{n!}$ is $\frac{n{z}^{n-1}}{n!}$. For $n=0$, the derivative is zero since $z^0$ is a constant.

Simplify the differentiated series

After differentiating, the series becomes:$$\sum _{n=1}^{\mathrm{\infty }}\frac{n{z}^{n-1}}{n!}=\sum _{n=1}^{\mathrm{\infty }}\frac{z^{n-1}}{(n-1)!}.$$This is obtained because $n/n!=1/(n-1)!$. By shifting the index of summation $m=n-1$, the series becomes:$$\sum _{m=0}^{\mathrm{\infty }}\frac{z^m}{m!}.$$

Recognize the resulting series as the original series

The series $\sum _{m=0}^{\mathrm{\infty }}\frac{z^m}{m!}$ is the original definition of the exponential function $e^z$. Therefore, the derivative of $e^z$ is $e^z$ itself.

Key concepts

Series Definition and Exponential Functions

The series definition is a powerful tool for understanding complex functions, such as the exponential function. In mathematics, an exponential function like $e^z$ can be expressed as an infinite series:

• $e^z=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{n!}$.

This representation allows us to break down the function into simpler pieces, making the function more approachable. By breaking down the exponential into a series, each term contributes a piece of the puzzle, where:

• $n!$ is the factorial of $n$.
• $z^n$ represents $z$ raised to the power of $n$.

Using such a series, we can perform operations like differentiation more straightforwardly, even with complex numbers. This definition provides a systematic way to expand and manipulate functions, crucial for solving mathematical problems.

Term-by-Term Differentiation of Series

When dealing with series like the one for the exponential function, term-by-term differentiation is a fundamental technique. This method involves differentiating each term of the series individually:

• Differentiate $\frac{z^n}{n!}$ to obtain $\frac{n{z}^{n-1}}{n!}$.
• For the first term when $n=0$, the derivative is zero because it's a constant.

The beauty of term-by-term differentiation lies in its simplicity and directness. It relies on basic calculus rules applied systematically:

• We don't differentiate the entire series at once, rather each term on its own.
• It's a method that can often simplify solving differential equations.

This process leads to a new series that, for exponential functions, interestingly resembles the original function itself. This property is what makes functions like $e^z$ so extraordinary and elegant in calculus and analysis.

Complex Analysis and Exponential Functions

Complex analysis introduces us to the fascinating world of functions of complex numbers. Here, the exponential function $e^z$ becomes even more interesting. In complex analysis:

• $z$ represents a complex number, often written as $z=x+yi$ where $x$ and $y$ are real numbers.
• The function $e^z$ extends its properties from real numbers to this complex plane.

An exponential function in complex analysis retains its identity under differentiation:

• The derivative of $e^z$ is $e^z$, maintaining its form even for complex numbers.

This property highlights the harmony and predictability of exponential functions, whether in real or complex domains. Complex analysis uses these properties to explore deeper relationships and solve more intricate problems, revealing patterns that are both elegant and powerful.