Euler transforms between complex and trigonometric expressions using the
identities in Figure
3.5. To used the transformation, provide a subject that matches one side
or the other of one of the equations.
(ⅇ^(0, x)ⅈ+ⅇ^(0, -x)ⅈ)÷2=(cos x)ⅈ
(ⅇ^(0, x)ⅈ-ⅇ^(0, -x)ⅈ)÷2ⅈ=(sin x)ⅈ
ⅇ^(0, x)ⅈ=(cos x, sin x)ⅈ
ⅇ^(0, -x)ⅈ=(cos x, -sin x)ⅈ
Figure 3.5 Euler identities
The expressions x and -x represent the imaginary components of a complex value.
The transformation provides considerable flexibility for this expression.
The exponents can be any matching complex values with zero real components and opposite signs in the
imaginary components. For example, ⅇ^(0, 2⋅ℼ⋅x)ⅈ+ⅇ^-(0, 2⋅ℼ⋅x)ⅈ transforms to 2⋅cos (2⋅ℼ⋅x).
The transformation also collects factors, divisors, signs and exponents before looking for matching exponents
of ⅇ. Applied to the integrand in ∫cos x^2 ⅆx, the transformation produces
∫((ⅇ^(0, x)ⅈ+ⅇ^(0, -x)ⅈ)÷2)^2 ⅆx. After distributing several times, this becomes
Applying the Euler transformation directly to the integrand produces ∫1/2⋅cos (2⋅x)+ⅇ^0÷4+ⅇ^0÷4 ⅆx
which integrates to 1/4⋅sin (2⋅x)+1/2⋅x.
The Euler transformation also works on the ⅉ form of complex expression. In this mode,
the expression sin (2⋅x)^2
transforms to ((ⅇ^(2⋅x⋅ⅉ)-ⅇ^(-(2⋅x)⋅ⅉ))÷(-2⋅ⅉ))^2. Distributing several time produces
(ⅇ^(2⋅x⋅ⅉ)-ⅇ^(-(2⋅x)⋅ⅉ))^2÷(-2⋅ⅉ)^2, (ⅇ^(2⋅x⋅ⅉ)-ⅇ^(-(2⋅x)⋅ⅉ))^2÷(-2⋅ⅉ)^2
and
With some work, this simplifies to -(ⅇ^(4⋅x⋅ⅉ)-2+ⅇ^(-4⋅x⋅ⅉ))÷4. Selecting the terms in
ⅇ, Euler transforms the expression to -(2⋅cos (4⋅x)-2)÷4. This simplifies to
-(1/2)⋅cos (4⋅x)+1/2.
The latter example bears some thought. How does Euler know to transform to the ⅈ form
or the ⅉ form of expression? There are no hints in the subject of the transformation.
User's of Myron will likely choose one form over the other for most of their work.
The Euler transformation uses the last
complex input as the mode for conversion to complex. The mode can be switched by applying Euler to
simple throw-away expressions like ⅈ and ⅉ to set the mode.
