Euler formula
Description of formula: In the formula, e is the base of natural logarithm and I is the imaginary unit.
e^(ix)=cosx+isinx
E is the base of natural logarithm, and I is the imaginary unit.
It extends the definition domain of trigonometric function to complex number, and establishes the relationship between trigonometric function and exponential function, which occupies a very important position in the theory of complex variable function.
Replace x in the formula with -x to get:
E (-ix) = cosx-isinx, and then we add and subtract the two formulas to get:
sinx=[e^(ix)-e^(-ix)]/(2i),cosx=[e^(ix)+e^(-ix)]/2.
These two are also called Euler formulas.
The formula created by God
Let x in e (ix) = cosx+isinx be π, and then:
e^(iπ)+ 1=0.
This equation, also known as Euler formula, is the most fascinating formula in mathematics. It connects several most important numbers in mathematics: two transcendental numbers: the base e and π of natural logarithm, two units: imaginary number unit I and natural number unit 1, and the common 0 in mathematics. Mathematicians evaluate it as a "formula created by God", which we can only look at but can't understand.