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Question Number 38112 by maxmathsup by imad last updated on 21/Jun/18

prove that  arctan(x)= (i/2)ln(((i+x)/(i−x))) for ∣x∣<1

$${prove}\:{that}\:\:{arctan}\left({x}\right)=\:\frac{{i}}{\mathrm{2}}{ln}\left(\frac{{i}+{x}}{{i}−{x}}\right)\:{for}\:\mid{x}\mid<\mathrm{1} \\ $$

Commented by prof Abdo imad last updated on 24/Jun/18

we have i+x=x+i=(√(x^2 +1))( (x/(√(x^2  +1))) +(i/(√(x^2  +1))))  =r e^(iθ)  ⇒r=(√(x^2 +1)) and cosθ=(x/(√(x^2  +1)))  sinθ=(1/(√(x^2 +1))) ⇒ tanθ=(1/x) ⇒θ=arctan((1/x)) ⇒  x+i =(√(x^2 +1)) e^(i arctan((1/x)))  ⇒  ln(x+i)=(1/2)ln(x^(2 ) +1) +iarctan((1/x))and  ln(−x+i)=ln(−1) +ln(x−i)  =iπ  +(1/2)ln(x^2 +1)−i arctan((1/x)) ⇒  (i/2)ln(((i+x)/(i−x)))=(i/2){ln(i+x)−ln(i−x)}  =(i/2){ 2i arctan((1/x))−iπ}  =(π/2) −arctan((1/x))=arctan(x) if 0<x<1  if −1<x<0 we use the chang.x=−t...

$${we}\:{have}\:{i}+{x}={x}+{i}=\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\left(\:\frac{{x}}{\sqrt{{x}^{\mathrm{2}} \:+\mathrm{1}}}\:+\frac{{i}}{\sqrt{{x}^{\mathrm{2}} \:+\mathrm{1}}}\right) \\ $$$$={r}\:{e}^{{i}\theta} \:\Rightarrow{r}=\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\:{and}\:{cos}\theta=\frac{{x}}{\sqrt{{x}^{\mathrm{2}} \:+\mathrm{1}}} \\ $$$${sin}\theta=\frac{\mathrm{1}}{\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}\:\Rightarrow\:{tan}\theta=\frac{\mathrm{1}}{{x}}\:\Rightarrow\theta={arctan}\left(\frac{\mathrm{1}}{{x}}\right)\:\Rightarrow \\ $$$${x}+{i}\:=\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\:{e}^{{i}\:{arctan}\left(\frac{\mathrm{1}}{{x}}\right)} \:\Rightarrow \\ $$$${ln}\left({x}+{i}\right)=\frac{\mathrm{1}}{\mathrm{2}}{ln}\left({x}^{\mathrm{2}\:} +\mathrm{1}\right)\:+{iarctan}\left(\frac{\mathrm{1}}{{x}}\right){and} \\ $$$${ln}\left(−{x}+{i}\right)={ln}\left(−\mathrm{1}\right)\:+{ln}\left({x}−{i}\right) \\ $$$$={i}\pi\:\:+\frac{\mathrm{1}}{\mathrm{2}}{ln}\left({x}^{\mathrm{2}} +\mathrm{1}\right)−{i}\:{arctan}\left(\frac{\mathrm{1}}{{x}}\right)\:\Rightarrow \\ $$$$\frac{{i}}{\mathrm{2}}{ln}\left(\frac{{i}+{x}}{{i}−{x}}\right)=\frac{{i}}{\mathrm{2}}\left\{{ln}\left({i}+{x}\right)−{ln}\left({i}−{x}\right)\right\} \\ $$$$=\frac{{i}}{\mathrm{2}}\left\{\:\mathrm{2}{i}\:{arctan}\left(\frac{\mathrm{1}}{{x}}\right)−{i}\pi\right\} \\ $$$$=\frac{\pi}{\mathrm{2}}\:−{arctan}\left(\frac{\mathrm{1}}{{x}}\right)={arctan}\left({x}\right)\:{if}\:\mathrm{0}<{x}<\mathrm{1} \\ $$$${if}\:−\mathrm{1}<{x}<\mathrm{0}\:{we}\:{use}\:{the}\:{chang}.{x}=−{t}... \\ $$$$ \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 22/Jun/18

ln(α+iβ)=(1/2)ln(α^2 +β^2 )+i(2nΠ+tan^(−1) (β/α))  ln(((i+x)/(i−x)))=ln(i+x)−ln(i−x)  ln(i+x)=(1/2)ln(1+x^2 )+i(2nΠ+tan^(−1) (1/x))  ln(i−x)=(1/2)ln(1+x^2 )+i(2nΠ+tan^(−1) (1/(−x)))  ln(((i+x)/(i−x)))=i(tan^(−1) (1/x)−tan^(−1) (1/(−x)))  =i(2tan^(−1) (1/x))  so the value  (i/2)ln(((i+x)/(i−x)))  (i/2)×i(2tan^(−1) (1/x))  =−tan^(−1) ((1/x))=tan^(−1) (x)−(Π/2)  since  tan^(−1) x=k  tank=x  cotk=(1/x)  tan((Π/2)−k)=(1/x)  tan^(−1) ((1/x))=(Π/2)−k  tan^(−1) ((1/x))+tan^(−1) (x)=(Π/2)

$${ln}\left(\alpha+{i}\beta\right)=\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} \right)+{i}\left(\mathrm{2}{n}\Pi+{tan}^{−\mathrm{1}} \frac{\beta}{\alpha}\right) \\ $$$${ln}\left(\frac{{i}+{x}}{{i}−{x}}\right)={ln}\left({i}+{x}\right)−{ln}\left({i}−{x}\right) \\ $$$${ln}\left({i}+{x}\right)=\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)+{i}\left(\mathrm{2}{n}\Pi+{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{{x}}\right) \\ $$$${ln}\left({i}−{x}\right)=\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)+{i}\left(\mathrm{2}{n}\Pi+{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{−{x}}\right) \\ $$$${ln}\left(\frac{{i}+{x}}{{i}−{x}}\right)={i}\left({tan}^{−\mathrm{1}} \frac{\mathrm{1}}{{x}}−{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{−{x}}\right) \\ $$$$={i}\left(\mathrm{2}{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{{x}}\right) \\ $$$${so}\:{the}\:{value} \\ $$$$\frac{{i}}{\mathrm{2}}{ln}\left(\frac{{i}+{x}}{{i}−{x}}\right) \\ $$$$\frac{{i}}{\mathrm{2}}×{i}\left(\mathrm{2}{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{{x}}\right) \\ $$$$=−{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{{x}}\right)={tan}^{−\mathrm{1}} \left({x}\right)−\frac{\Pi}{\mathrm{2}} \\ $$$${since} \\ $$$${tan}^{−\mathrm{1}} {x}={k} \\ $$$${tank}={x} \\ $$$${cotk}=\frac{\mathrm{1}}{{x}} \\ $$$${tan}\left(\frac{\Pi}{\mathrm{2}}−{k}\right)=\frac{\mathrm{1}}{{x}} \\ $$$${tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{{x}}\right)=\frac{\Pi}{\mathrm{2}}−{k} \\ $$$${tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{{x}}\right)+{tan}^{−\mathrm{1}} \left({x}\right)=\frac{\Pi}{\mathrm{2}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

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