The concept of logarithms extends naturally from real numbers to complex numbers, but it requires careful consideration due to the multi-valued nature of complex exponentiation. Understanding how to derive the value of $\log(-1)$ within the framework of complex analysis reveals the intricate relationship between logarithms, exponentials, and the complex plane.
The Complex Exponential and Logarithm
In complex analysis, the exponential function is defined as:
$e^z = e^{x + iy} = e^x (\cos y + i \sin y)$
where $z = x + iy$ is a complex number, and $i$ is the imaginary unit ($i^2 = -1$).
The complex logarithm is the inverse of the complex exponential function. In other words, if $w = e^z$, then $z = \log w$. However, unlike the real logarithm, the complex logarithm is multi-valued. This arises from the periodic nature of the trigonometric functions within the complex exponential.
For example, consider the equation $e^{2 \pi i} = 1$. Since the exponential function is periodic with period $2 \pi i$, we have $e^{2 \pi i} = e^{4 \pi i} = e^{6 \pi i} = \dots = 1$. This means that the logarithm of 1 is not a single value but rather a set of values:
$\log 1 = 0, 2\pi i, 4 \pi i, 6 \pi i, \dots$
Deriving the Value of $\log(-1)$
To find the value of $\log(-1)$, we seek a complex number $z$ such that $e^z = -1$. We can express $-1$ in polar form as $-1 = e^{\pi i}$. Using the definition of the complex exponential, we get:
$e^z = e^{x + iy} = e^x (\cos y + i \sin y) = -1 = e^{\pi i}.$
Matching the real and imaginary parts, we have:
$e^x \cos y = -1$ $e^x \sin y = 0$
From the second equation, we see that $\sin y = 0$, which implies $y = k \pi$ where $k$ is an integer. Since we want $e^x \cos y = -1$, we need $\cos y = -1$, which occurs when $y = (2k + 1) \pi$ for any integer $k$.
Therefore, the general solution for $z$ is:
$z = x + iy = x + (2k + 1) \pi i$
where $x$ is any real number. This implies that $\log(-1)$ has infinitely many values, each differing by a multiple of $2\pi i$. We can express this as:
$\log(-1) = \boxed{(2k + 1)\pi i}, \text{ where } k \in \mathbb{Z}.$
The Principal Value of $\log(-1)$
To avoid ambiguity, we often define a principal value of the complex logarithm, denoted by $\text{Log } z$, which is the unique value of $\log z$ with the imaginary part lying in the interval $(-\pi, \pi]$.
In the case of $\log(-1)$, the principal value is:
$\text{Log } (-1) = \pi i.$
This value corresponds to $k = 0$ in the general solution for $\log(-1)$.
Conclusion
Deriving the value of $\log(-1)$ in the realm of complex analysis highlights the multi-valued nature of complex logarithms. While there are infinitely many possible values for $\log(-1)$, each differing by a multiple of $2\pi i$, the principal value, $\pi i$, provides a unique and consistent representation within the framework of complex analysis. Understanding the complex logarithm's properties is crucial for navigating complex function theory, including power series, integration, and the study of holomorphic functions. This fundamental concept has far-reaching implications in diverse scientific fields, from quantum mechanics and signal processing to electrical engineering and fluid dynamics.
