Mathematical Methods for Physics and Engineering
1. Complex number division [Page 90-91]:
\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}
2. Complex number logarithms [Page 100]:
z=re^{i(\theta+2n\pi)},\ Lnz=lnr+i(\theta+2n\pi)
3. Complex number applied to differentiation and integration [Page 101]:
Sometimes complementing trigonometric functions to be complex functions will simplify the differentiation or integration.
Example:
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
4. Hyperbolic functions [Page 102-108]:
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)\\
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