Hyperbolic Functions & Logarithms of Complex Numbers
- How are sin z and cos z defined when z is a complex number, and why can they exceed 1? In one line For complex z , sin z=dfrace^iz-e^(-iz)2i and cos z=dfrace^iz+e^(-iz)2 ; these come from Euler's formula and, unlike the real ca... 1
- What are hyperbolic functions like sinh x, cosh x and tanh x, and how are they defined? In one line Hyperbolic functions are built from e^x : sinh x=dfrace^x-e^(-x)2 , cosh x=dfrace^x+e^(-x)2 , tanh x=sinh x/cosh x ; they behave... 1
- How are circular and hyperbolic functions related, for example why is sin(ix) = i sinh x? In one line Putting an imaginary argument into a circular function gives a hyperbolic function and vice versa: sin ix=isinh x , cos ix=cosh... 1
- How do I separate sin(x+iy), cos(x+iy), sinh(x+iy) and cosh(x+iy) into real and imaginary parts? In one line Expand with the addition formula and then use cos iy=cosh y , sin iy=isinh y ; for example sin(x+iy)=sin xcosh y+icos xsinh y .... 1
- What are the main hyperbolic identities and how does Osborn's rule help me remember them? In one line Every trigonometric identity has a hyperbolic twin, obtained by changing cos→cosh , sin→sinh and flipping the sign of every term... 1
- What is the logarithm of a complex number and what are its general and principal values? In one line For z=re^iθ , log z=log r+i(θ+2nπ) ; the principal value takes n=0 and θ as the principal argument. Explained simply For positiv... 1
- How do I separate tan(x+iy) into real and imaginary parts and solve problems based on it? In one line Multiply numerator and denominator by the conjugate cos(x-iy) to get tan(x+iy)=sin2x+isinh2y/cos2x+cosh2y . Explained simply For... 1
- How do I find real and imaginary parts of complex powers like i^i and (1+i)^(1−i)? In one line Use a^b=e^(blog a) : find log a for the complex base, multiply by the exponent, separate e^(u+iv)=e^u(cos v+isin v) . Explained... 1
- How do I express inverse hyperbolic functions like sinh⁻¹x, cosh⁻¹x and tanh⁻¹x as logarithms? In one line Because hyperbolic functions are built from e^x , their inverses are logarithms: sinh⁻¹x=log(x+√(x²+1)) , cosh⁻¹x=log(x+√(x²-1))... 1
- How are complex numbers used to analyse AC electrical circuits with impedance and phasors? In one line In AC circuits a sinusoidal voltage or current is represented by a complex number (phasor), and each component by a complex impe... 1
- How do I find real and imaginary parts of inverse circular functions like tan⁻¹(α + iβ)? In one line Write the inverse function as an ordinary equation, e.g. tan⁻¹(α+iβ)=x+iy means tan(x+iy)=α+iβ , then use the conjugate and sepa... 1
- If \(\alpha+i\beta=\tanh\left(x+\frac{i\pi}{4}\right)\), prove that \(\alpha^2+\beta^2=1\). Separate real and imaginary parts: tanh(x+iy)=sinh(x+iy)cosh(x-iy)/cosh(x+iy)cosh(x-iy) . Numerator =1/2[sinh2x+sinh2iy]=1/2[sinh2x+isin2y]... 1
- If \(5\sinh x-\cosh x=5\), find \(\tanh x\). 5·frace^x-e^(-x)2-frace^x+e^(-x)2=5⇒4e^x-6e^(-x)=10 . Multiply by e^x and put e^x=t : 4t²-10t-6=0⇒2t²-5t-3=0⇒(2t+1)(t-3)=0 . t=e^x>0⇒ e^x=3... 1
- If \(\tan[\log(x+iy)]=a+ib\), prove that \(\tan[\log(x^2+y^2)]=\frac{2a}{1-a^2-b^2}\) when \(a^2+b^2\ne1\). Let log(x+iy)=α+iβ . Then α=log√(x²+y²)=1/2log(x²+y²) , so log(x²+y²)=2α . Given tan(α+iβ)=a+ib ; taking conjugates, tan(α-iβ)=a-ib . tan2α=... 1
- Prove that \(\log\left(\frac{x+iy}{x-iy}\right)=2i\tan^{-1}\left(\frac yx\right)\). Let x+iy=re^iθ where r=√(x²+y²),\ θ=tan⁻¹y/x . Then x-iy=re^(-iθ) . log(x+iy/x-iy)=logfracre^iθre^(-iθ)=log e^2iθ=2iθ (principal value) =2it... 1
- If \(\tan(\alpha+i\beta)=e^{i\theta}\), prove that \(\alpha=\frac{n\pi}2+\frac\pi4\) and \(\beta=\frac12\log\left[\tan\left(\frac\pi4+\frac\theta2\right)\right]\). tan(α+iβ)=cosθ+isinθ⇒tan(α-iβ)=cosθ-isinθ . For α : tan2α=tan[(α+iβ)+(α-iβ)]=2cosθ/1-(cos²θ+sin²θ)=2cosθ/0=∞ . ⇒2α=nπ+fracπ2⇒ α=fracnπ2+frac... 1
- Show that \(\log[\sin(x+iy)]=\frac12\log\left(\frac{\cosh2y-\cos2x}2\right)+i\tan^{-1}(\cot x\tanh y)\). sin(x+iy)=sin xcos iy+cos xsin iy=sin xcosh y+icos xsinh y . For log(a+ib)=1/2log(a²+b²)+itan⁻¹b/a : a²+b²=sin²xcosh²y+cos²xsinh²y=sin²x(1+s... 1
- Prove that \(\log\left(e^{i\alpha}+e^{i\beta}\right)=\log\left[2\cos\left(\frac{\alpha-\beta}{2}\right)\right]+i\left(\frac{\alpha+\beta}{2}\right)\). Take out e^(i(α+β)/2) : e^iα+e^iβ=e^(iα+β/2)(e^(iα-β/2)+e^(-iα-β/2))=e^(iα+β/2)·2cosα-β/2 (using e^iθ+e^(-iθ)=2cosθ ). Taking logarithm (pri... 1
- Prove that \(\mathrm{sech}^{-1}(\sin\theta)=\log\left(\cot\frac{\theta}{2}\right)\). Let sech⁻¹(sinθ)=u . Then sech u=sinθ⇒cosh u=1/sinθ (take 0<θ<π ). sinh u=√(cosh²u-1)=√(1-sin²θ/sin²θ)=cosθ/sinθ . e^u=cosh u+sinh u=1+cosθ/... 1
- Prove that: \(\tanh(\log\sqrt3)=0.5\) tanh x=frace^x-e^(-x)e^x+e^(-x) . With x=logsqrt3 : e^x=sqrt3,\ e^(-x)=1/sqrt3 . tanh(logsqrt3)=fracsqrt3-frac1sqrt3sqrt3+frac1sqrt3=3-1/3+1... 1