# Mathematician:Johann Peter Gustav Lejeune Dirichlet

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## Mathematician

German mathematician who worked mainly in the field of analysis.

Credited with the first formal definition of a function.

Married Rebecka Mendelssohn, the youngest sister of Fanny Mendelssohn and Felix Mendelssohn.

## Nationality

German, although born in what was then part of the French empire.

## History

- Born: 13 Feb 1805, Düren, French Empire (now Germany)
- 1855: Succeeded Gauss in professorship at Göttingen.
- Died: 5 May 1859, Göttingen, Hanover (now Germany)

## Theorems and Definitions

### Definitions

- Dirichlet Tessellation (also known as a Voronoi Diagram) and hence:

- Dirichlet Character in Number Theory, specifically:

- Dirichlet Conditions (for Fourier Series)
- Dirichlet Convolution (Number Theory and Arithmetic Functions)
- Dirichlet Density (Number Theory)
- Dirichlet Distribution (Probability Theory)
- Dirichlet Eta Function (Number Theory)
- Dirichlet Form
- Dirichlet Function
- Modified Dirichlet Function (also known as the Thomae Function or small Riemann function)

- Dirichlet Kernel (Functional Analysis, Fourier Series)
- Latent Dirichlet Allocation (Statistics)

Definitions of concepts named for **Johann Peter Gustav Lejeune Dirichlet** can be found here.

### Theorems

- Theorems named Dirichlet's Theorem:
- Dirichlet's Approximation Theorem (Diophantine Equations)
- Dirichlet's Theorem on Arithmetic Sequences (Number Theory, specifically prime numbers)
- Dirichlet's Unit Theorem (Algebraic Number Theory and Ring Theory)
- Dirichlet's Theorem for 1-Dimensional Fourier Series, also known as Fourier's Theorem, for Joseph Fourier

- Dirichlet Problem (Partial Differential Equations)
- Dirichlet Stability Criterion (Dynamical Systems Theory)
- Dirichlet's Test for Uniform Convergence (Analysis)
- Dirichlet Boundary Condition (Differential Equations)

- Dirichlet Divisor Problem (currently unsolved) (Number Theory)
- Class Number Formula (Analysis)
- Dirichlet Integral (Integral Calculus)

Results named for **Johann Peter Gustav Lejeune Dirichlet** can be found here.

## Publications

## Critical View

*The story was told that young Dirichlet had as a constant companion on all his travels, like a devout man with his prayer book, an old, worn copy of the**Disquisitiones Arithmeticae*of Gauss.

*Dirichlet was not satisfied to study Gauss's*Disquisitiones*once or several times, but continued throughout his life to keep in close touch with the wealth of deep mathematical thoughts which it contains by perusing it again and again. For this reason the book was never put on the shelf but had an abiding place on the table at which he worked. Dirichlet was the first one who not only fully understood this work, but also made it accessible to others.*

## Also known as

Usually referred to as **Peter Dirichlet**.

The **Johann** is often ignored or forgotten even when specifying his name more fully: **Peter Gustav Lejeune Dirichlet**.

## Sources

- John J. O'Connor and Edmund F. Robertson: "Johann Peter Gustav Lejeune Dirichlet": MacTutor History of Mathematics archive

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**Dirichlet, Peter Gustav Lejeune**(1805-59) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.28$: Dirichlet ($\text {1805}$ – $\text {1859}$) - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Dirichlet, Peter Gustav Lejeune**(1805-59) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Dirichlet, Peter Gustav Lejeune**(1805-59) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Dirichlet, Peter Gustav Lejeune**(1805-59)