Ch3-Complex numbers and hyperbolic functions-CHINKEII's notes

Mathematical Methods for Physics and Engineering

\frac{z_1}{z_2}=\frac{x_1x_2+y_1y_2}{x_2^2+y_2^2}+i\frac{x_2y_1-x_1y_2}{x_2^2+y_2^2}

Especially for conjugate division:

\frac{z}{z^*}=\frac{x^2-y^2}{x^2+y^2}+i\frac{2xy}{x^2+y^2}

z=re^{i(\theta+2n\pi)},\ Lnz=lnr+i(\theta+2n\pi)

Sometimes complementing trigonometric functions to be complex functions will simplify the differentiation or integration.

Evaluate the integral $I=\int e^{ax}cosbxdx$ .

I=\int e^{ax}cosbxdx=Re[\int e^{(a+ib)x}dx]\\ =Re[\frac{e^{ax}}{a^2+b^2}(ae^{ibx}-ibe^{ibx})+c]\\ =\frac{e^{ax}}{a^2+b^2}(acosbx+bsinbx)+c_1

Definition:

sinhx=\frac{e^x-e^{-x}}{2},\ coshx=\frac{e^x+e^{-x}}{2}\\ tanhx=\frac{e^x-e^{-x}}{e^x+e^{-x}},\ cothx=\frac{e^x+e^{-x}}{e^x-e^{-x}}\\ sechx=\frac{2}{e^x+e^{-x}},\ cosechx=\frac{2}{e^x-e^{-x}}\\

Relations between hyperbolic and trigonometric functions:

cosh(x)=cos(ix),\ cos(x)=cosh(ix)\\ isinh(x)=sin(ix),\ isin(x)=sinh(ix)

Identities of hyperbolic functions:

sech^2x=1-tanh^2x\leftrightarrow sec^2x=1+tan^2x\\ cosech^2x=coth^2x-1\leftrightarrow cosec^2x=cot^2x+1\\ sinh2x=2sinhxcoshx\leftrightarrow sin2x=2sinxcosx\\ cosh2x=cosh^2x+sinh^2x\leftrightarrow cos2x=cos^2x-sin^2x

Inverses of hyperbolic functions:

sinh^{-1}x=ln(\sqrt{x^2+1}+x)\\ cosh^{-1}x=ln(\sqrt{x^2-1}+x)\\ tanh^{-1}x=\frac{1}{2}ln(\frac{1+x}{1-x})

Differentiation of hyperbolic functions:

\frac{d}{dx}sinhx=coshx,\ \frac{d}{dx}coshx=sinhx\\ \frac{d}{dx}tanhx=sech^2x,\ \frac{d}{dx}sechx=-sechxtanhx\\ \frac{d}{dx}cosechx=-cosechxcothx,\ \frac{d}{dx}cothx=-cosech^2x\\ \frac{d}{dx}(sinh^{-1}\frac{x}{a})=\frac{1}{\sqrt{x^2+a^2}}\\ \frac{d}{dx}(cosh^{-1}\frac{x}{a})=\frac{1}{\sqrt{x^2-a^2}}\\ \frac{d}{dx}(tanh^{-1}\frac{x}{a})=\frac{a}{a^2-x^2},(x^2<a^2)\\ \frac{d}{dx}(coth^{-1}\frac{x}{a})=\frac{-a}{x^2-a^2},(x^2>a^2)\\