Problem 23 - Complex Analysis

Question

There is a problem posed by TH. CLAUSEN in Journal für reine and angewandte Mathematik (CRELLE's Journal), Band 2 (1827), pages 286-287: "If \(e\) is the base for the hyperbolic (= natural) logarithms, \(\pi\) denotes half of the perimeter of the unit circle and \(n\) is a positive or negative number, then it is well-known that $$ \begin{gathered} e^{2 n \pi \sqrt{-1}}=1 \\ e^{1+2 n \pi \sqrt{-1}}=e \end{gathered} $$ and thus also that $$ e^{(1+2 n \pi \sqrt{-1})^{2}}=e=e^{1+4 n \pi \sqrt{-1}-4 n^{2} \pi^{2}} $$ However, since \(e^{1+4 n \pi \sqrt{-1}}=e\), it would follow from this that \(e^{-4 n^{2} \pi^{2}}=1\), which is absurd. Find the mistake in the derivation of this result."

Step-by-Step Solution

Answer

The mistake is assuming \( e^{-4n^2\pi^2} = 1 \). It should be \( e^{-4n^2\pi^2} < 1 \) for non-zero \( n \).

Step 1: Understanding Exponential Equations

The expressions given involve the properties of complex exponentials. Recall that, for any integer \( n \), \( e^{2 n \pi \sqrt{-1}} = 1 \) because \( e^{ix} = \cos(x) + i\sin(x) \) and \( \cos(2n\pi) + i\sin(2n\pi) = 1 \). Similarly, \( e^{1 + 2n\pi \sqrt{-1}} = e^1 \cdot e^{2n\pi \sqrt{-1}} = e \cdot 1 = e \). These results confirm the expression up to this point.

Step 2: Squaring the Complex Exponent

Consider the equation \( e^{(1 + 2n\pi \sqrt{-1})^2} = e \). Expanding the square, we get \((1 + 2n\pi \sqrt{-1})^2 = 1 + 4n\pi \sqrt{-1} - 4n^2\pi^2\). So, the equation becomes \( e^{1 + 4n\pi \sqrt{-1} - 4n^2\pi^2} \). The claim that this is equal to \( e \) comes from assuming \( e^{1 + 4n\pi \sqrt{-1}} = e \), which is correct as established before, and thus assumes \( e^{-4n^2\pi^2} = 1 \).

Step 3: Analyzing the Exponential Property Mistake

The mistake arises because \( e^{-4n^2\pi^2} eq 1 \) for non-zero \( n \). The argument incorrectly assumes that the imaginary part of the exponent cancels out the real part. The property \( e^{a+bi} = e^a \cdot e^{bi} \) separates but does not imply individual terms are equal to 1 unless \( a = 0 \). Here, the exponent \(-4n^2\pi^2\) is real and very negative for any non-zero \( n \), indicating \( e^{-4n^2\pi^2} \) is a positive number much less than 1.

Key Concepts

Complex Numbers

Complex numbers are an extension of the real numbers and are used in various areas of mathematics including solving equations and modeling oscillations. A complex number is written in the form:

• \( z = a + bi \)

where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit satisfying \( i^2 = -1 \).
In practice, complex numbers allow us to solve equations that have no real solutions, such as \( x^2 + 1 = 0 \). Using complex numbers, any polynomial equation has a solution, a concept that underpins much of algebra.
A complex number can also be represented geometrically on a two-dimensional plane, known as the complex plane, with the horizontal axis as the real part and the vertical axis as the imaginary part. This representation helps us easily visualize operations like addition, subtraction, and multiplication of complex numbers.

Exponential Functions

The exponential function is a fundamental mathematical function that has the form \( e^x \), where \( e \) is a constant approximately equal to 2.71828. This function is unique because it is the only function that is its own derivative, meaning that

• \( \frac{d}{dx}e^x = e^x \).

Exponential functions model growth processes like population growth, radioactive decay, and compound interest.
In the context of complex numbers, exponential functions take the form \( e^{a+bi} \). Using Euler's formula, \( e^{bi} = \cos(b) + i\sin(b) \), the function can express oscillations as a combination of sine and cosine functions. This means exponentials of complex numbers can represent physical phenomena such as waves and electric circuits.
In examining the expressions under study, the exponential's periodicity \( e^{2n\pi i} = 1 \) shows how multiplying by an integral number of \( 2\pi \) wraps around the unit circle in the complex plane, leading back to the starting point, demonstrating a fundamental property of complex exponentials.

Mathematical Error Analysis

Error analysis in mathematics is crucial for identifying and understanding mistakes in derivations and problem solutions. In this context, an error occurred because of a misinterpretation of the properties of exponential functions with complex numbers.
The mistake was assuming that

• \( e^{-4n^2\pi^2} = 1 \)

for non-zero \( n \).
The error analysis involves carefully inspecting the separation of real and imaginary components. For an expression \( e^{a+bi} = e^a \cdot e^{bi} \), each portion affects the magnitude and direction in the complex plane differently. The real component \( e^a \) affects the magnitude, especially when negative, making it quite small rather than equal to 1.
In this case, for any non-zero \( n \), the presence of the negative real term \( -4n^2\pi^2 \) leads to an exponential value that is small (much less than 1), debunking the false assumption and highlighting the importance of treating each part separately to prevent incorrect conclusions.