Separation of Real and Imaginary Parts Formulas

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1. Separation of Algebraic Functions

If the given expression is of the form (x + iy)ⁿ, then for its separation into real and imaginary parts, we make the following substitutions in it x = r cos θ, y = r sin θ
(x + iy)ⁿ = rⁿ (cos θ + i sin θ)ⁿ = rⁿ (cos nθ + i sin nθ)
If the value of n is less then use Binomial theorem and when n is more then use De-moiver’s theorem.

2. Separation of Inverse Trigonometric & Inverse Hyperbolic functions

(i) tan^-1 (x + iy) = \(\frac{1}{2} \tan ^{-1}\left(\frac{2 x}{1-x^{2}-y^{2}}\right)+\frac{i}{2} \tanh ^{-1}\left(\frac{2 y}{1+x^{2}+y^{2}}\right)\)
or
\(\frac{1}{2} \tan ^{-1}\left(\frac{2 \mathrm{x}}{1-\mathrm{x}^{2}-\mathrm{y}^{2}}\right)+\frac{\mathrm{i}}{4} \log \left[\frac{\mathrm{x}^{2}+(1+\mathrm{y})^{2}}{\mathrm{x}^{2}+(1-\mathrm{y})^{2}}\right]\)

(ii) sin^-1(cosθ + i sinθ) = cos^-1 (\(\sqrt{\sin \theta}\)) + i sinh^-1 (\(\sqrt{\sin \theta}\))
or
= cos^-1 (\(\sqrt{\sin \theta}\)) + i log (\(\sqrt{\sin \theta}\) + \(\sqrt{1+\sin \theta}\))

(iii) cos^-1 (cos θ + i sin θ) = sin^-1 (\(\sqrt{\sin \theta}\)) – i sinh^-1 (\(\sqrt{\sin \theta}\))
or
= sin^-1 (\(\sqrt{\sin \theta}\)) – i log (\(\sqrt{\sin \theta}\) + \(\sqrt{1+\sin \theta}\))

(iv) tan^-1 (cos θ + i sin θ) = \(\frac{\pi}{4}+\frac{i}{4}\) log \(\left(\frac{1+\sin \theta}{1-\sin \theta}\right)\), (cosθ) > 0 and
tan^-1 (cos θ + i sin θ) = \(\left(-\frac{\pi}{4}\right)+\frac{i}{4} \log \left(\frac{1+\sin \theta}{1-\sin \theta}\right)\), (cosθ) < 0

3. Separation of Trigonometric and Hyperbolic Functions

If the given expressions involve trigonometric or hyperbolic function, then we use the formulae of sin (x + iy), cos (x + iy) etc. which are given in the chapter “Hyperbolic Function”. For hyperbolic functions, we should first express them in the form of trigonometrical functions.

4. Separation of Logarithmic Functions

(i) Principal value
Log (x + iy) = \(\frac{1}{2}\) log (x² + y²) + i tan^-1 (y/x)

(ii) General value
Log (x + iy) = \(\frac{1}{2}\) log (x² + y²) + i [2nπ + tan^-1 (y/x)]

5. Separation of Exponential Functions

(i) e^x+iy = e^x. e^iy = e^x (cos y + i sin y)
Real part = e^x cos y and Imaginary part = e^x sin y

(ii) If a ∈ R, then
a^x+iy = a^x.a^iy = a^x. e^iyloga
a^x+iy = a^x [cos(y log a) + i sin(y log a)]

6. Separation of expression of the form (Function)^function

To find two parts of (α + iβ)^x+iy, first we change its base to e and reduce it to the forme^p+iq. Thus
(α + iβ)^x+iy = e^{(x+iy)log(α + iβ)}
\(e^{(x+i y)\left[\frac{1}{2} \log \left(\alpha^{2}+\beta^{2}\right)+2 n \pi i+i \tan ^{-1}\left(\frac{\beta}{\alpha}\right)\right]}\)
= e^p+iq
Where
p = \(\frac{x}{2}\) log (α² + β²) – y[2nπ + tan^-1\(\frac{β}{α}\) &
q = \(\frac{y}{2}\) log (α² + β²) + x[2nπ + tan^-1\(\frac{β}{α}\)

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