Mathematical notations that he introduced:
- Concept of a function (first to write f(x) to denote the function f applied to the argument x)^[16]
- Letter Σ for summations
- Letter $i$ to denote the imaginary unit
- Modern notation for the trigonometric functions
- Natural logarithm (now also known as Euler's number)
Differential equations
Defined logarithms for negative and complex numbers
Defined the exponential function for complex numbers and discovered its relation to the trigonometric functions
Development of power series, the expression of functions as sums of infinitely many terms^[17]
Euler–Bernoulli beam equation, a cornerstone of engineering
Euler's critical load, the critical buckling load of an ideal strut
Euler equations in Fluid dynamics
Euler's formula $e^{i\varphi }=\cos \varphi +i\sin \varphi$
Euler's identity $e^{i\pi }+1=0$
Introduction of exponential function and logarithms in analytic proofs
Predicted the phenomenon of cavitation in 1754 before observing it
Solution to the Basel problem (1735)

$y'=p(x)y+q(x)y^{n}.$

$\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}$

Daniel Bernoulli's contributions:

Gabriel Cramer
- Cramer's theorem (algebraic curves)
- In 1750 he published Cramer's rule, giving a general formula for the solution for any unknown in a linear equation system having a unique solution, in terms of determinants implied by the system. This rule is still standard.

Astronomy

Biology