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Euler's formula

Euler's formula establishes the relation between the complex exponential function and trigonometric functions

 


Explanation

Substitution of x = ix in the power series for the exponential function gives

The imaginary unit can be substituted to obtain

In brackets are the power series for the cosine and the sine, so that

Substitution of x = −ix in the power series for the exponential function gives

so

 


Additional calculations

We add both formulas

and subtract them also

These are the sine and the cosine expressed with complex exponential functions.

 


Additional information

The Swiss mathematician Leonhard Euler described this formula in 1748.

 


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