Proof of Euler's Relation

PROOF OF EULER’S RELATION
© Evan and Craig Lewis

Eulers Relation can be derived from Taylors Series, defined as:

Copywrite Evan & Craig Lewis

Brook Taylor 1685 - 1731
note: represents the k^th differential of the funtion f

also:

So:
(i)

and
(ii)

Notice that sin(x) has all odd denominators, while cos(x) has all even denominators. The series can be combined to create something that looks similar to Taylors Series of

:
(iii)

is the only mathmatical expression that is equal to it’s own differential.
Definition of Eulers Relation:

Where

Note the following equalities: (iv)

Interestingly:

and for all positive integer values of n:

PROOF

Substitute ix into the true equation shown in (iii):

see (iv)

Factor:

Replace first term with equation (ii) and second term with equation (i):

(v)

Similarly:

(vi)

Add (v) and (vi) to eliminate the

terms

(vii)
or subtract (v) from (vi) to eliminate the

terms:

(viii)

These equations can be substituted back into Eulers Relation (v) showing internal consistency:

i’s cancel

and

cancel

Proof of:

Substitute

with

in Eulers Relation (v)

Rearranging:

see (iii)

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