# Separation of Real and Imaginary Parts Formulas

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## List of Separation of Real and Imaginary Parts Formulae

If you are working on Separation of Real and Imaginary Parts of a Complex Number you can always seek help using the Separation of Real and Imaginary Parts Formulas provided. Furthermore, you will get the real and imaginary parts of a complex number in a matter of seconds after applying the Formulae. Try recalling the Separation of Real and Imaginary Parts Formula List to solve your problems on Complex Numbers effortlessly.

**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} = r^{n} (cos θ + i sin θ)^{n} = r^{n} (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 (x^{2} + y^{2}) + i tan^{-1} (y/x)

(ii) General value

Log (x + iy) = \(\frac{1}{2}\) log (x^{2} + y^{2}) + 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 (α^{2} + β^{2}) – y[2nπ + tan^{-1}\(\frac{β}{α}\) &

q = \(\frac{y}{2}\) log (α^{2} + β^{2}) + x[2nπ + tan^{-1}\(\frac{β}{α}\)