Separation of Real and Imaginary Parts Formulas

Separation of Real and Imaginary Parts Formulas

Worried on how to separate Real and Imaginary Parts of a Complex Number? Fear not! as you can make use of the Separation of Real and Imaginary Parts Formulas List provided to make your job simple. You will get the Real and Imaginary Parts of a complex number easily by taking the help of the Formulae provided here. Understand the concept much better and get the Separation of Real and Imaginary Parts at a faster pace by applying the Formulas.

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 = 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) ex+iy = ex. eiy = ex (cos y + i sin y)
Real part = ex cos y and Imaginary part = ex sin y

(ii) If a ∈ R, then
ax+iy = ax.aiy = ax. eiyloga
ax+iy = ax [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 formep+iq. Thus
(α + iβ)x+iy = e(x+iy)log(α + iβ)
\(e^{(x+i y)\left[\frac{1}{2} \log \left(\alpha^{2}+\beta^{2}\right)+2 n \pi i+i \tan ^{-1}\left(\frac{\beta}{\alpha}\right)\right]}\)
= ep+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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