Bernoulli Numbers in Mathematics

Bernoulli numbers are a sequence of rational numbers that play an important role in number theory, algebra, and calculus. They are named after Swiss mathematician Jakob Bernoulli (1655-1705) who introduced them in 1713 in his book "Ars Conjectandi".

Definition and Properties

Bernoulli numbers are defined by their generating function:

$\frac{x}{e^x-1}=\sum_{n=0}^\infty \frac{B_n}{n!}x^n$

where $B_n$ denotes the $n$-th Bernoulli number.

The first few Bernoulli numbers are:

$B_0=1, B_1=-\frac{1}{2}, B_2=\frac{1}{6}, B_3=0, B_4=-\frac{1}{30}, \ldots$

Some important properties of Bernoulli numbers include:

• Odd-indexed Bernoulli numbers are zero: $B_{2n+1}=0$ for $n\geq 1$.
• The second Bernoulli number is negative: $B_1=-\frac{1}{2}$.
• The Bernoulli numbers alternate in sign: $B_{2n}$ is positive for even $n$ and negative for odd $n$.
• The denominators of the Bernoulli numbers grow rapidly: $B_n$ has denominator at least $2n$.
• The Bernoulli numbers satisfy a recurrence relation: $B_n=1-\sum_{k=0}^{n-1}\binom{n}{k}\frac{B_k}{n-k+1}$.

Applications

Bernoulli numbers have many applications in mathematics, including:

• Number theory: Bernoulli numbers are related to the Riemann zeta function and the distribution of prime numbers.
• Algebraic geometry: Bernoulli numbers appear in the study of algebraic curves and surfaces.
• Calculus: Bernoulli numbers are used in the study of special functions such as the Gamma function and the Riemann zeta function.
• Physics: Bernoulli numbers appear in the study of fluid dynamics and the propagation of waves.

Conclusion

Bernoulli numbers are an important and fascinating sequence of rational numbers that have numerous applications in mathematics and physics. They are a testament to the power and beauty of mathematical ideas, and their study continues to inspire new discoveries and insights.

Links

ベルヌーイ数[JA]