Hyperbolic Functions & Logarithms of Complex Numbers
Notes
See all →Top questions
See all →- 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