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Question Number 73697 by mathmax by abdo last updated on 14/Nov/19

f i n d a f o r m u l a e f o r c a l c u l u s o f a r c t a n (x + i y)

Commented by mathmax by abdo last updated on 15/Nov/19

w e h a v e p r o v e d t h a t a r c t a n z = \frac{1}{2 i} l n (\frac{1 + i z}{1 - i z}) \Rightarrow

l n (x + i y) = \frac{1}{2 i} l n (\frac{1 + i (x + i y)}{1 - i (x + i y)}) = \frac{1}{2 i} l n (\frac{1 - y + i x}{1 + y - i x}) w e h a v e

∣ 1 - y + i x ∣= \sqrt{{(1 - y)}^{2} + x^{2}} \Rightarrow 1 - y + i x = \sqrt{{(1 - y)}^{2} + x^{2}} e^{i a r c t a n (\frac{x}{1 - y})}

∣ 1 + y - i x ∣= \sqrt{{(1 + y)}^{2} + x^{2}} \Rightarrow 1 + y - i x = \sqrt{{(1 + y)}^{2} + x^{2}} e^{- i a r c t a n (\frac{x}{1 + y})}

\Rightarrow a r c t a n (x + i y) = \frac{1}{2 i} l n (\frac{\sqrt{x^{2} + {(1 - y)}^{2}} e^{i a r c t a n (\frac{x}{1 - y})}}{\sqrt{x^{2} + {(y + 1)}^{2}} e^{- i a r c t a n (\frac{x}{1 + y})}})

= \frac{1}{4 i} l n (x^{2} + {(1 - y)}^{2}) + \frac{1}{2} a r c t a n (\frac{x}{1 - y}) - \frac{1}{4 i} l n (x^{2} + {(y + 1)}^{2})

+ \frac{1}{2} a r c t a n (\frac{x}{1 + y})

= - \frac{i}{4} l n (x^{2} + {(1 - y)}^{2}) + \frac{i}{4} l n (x^{2} + {(y + 1)}^{2}) + \frac{1}{2} a r c t a n (\frac{x}{1 - y})

+ \frac{1}{2} a r c t a n (\frac{x}{1 + y})

= \frac{1}{2} a r c t a n (\frac{x}{1 + y}) + \frac{1}{2} a r c t a n (\frac{x}{1 - y)}) + \frac{i}{4} l n (\frac{x^{2} + {(y + 1)}^{2}}{x^{2} + {(1 - y)}^{2}})

Answered by Smail last updated on 15/Nov/19

a r c t a n (x) = \frac{1}{2 i} l n (\frac{x - i}{x + i})

s o a r c t a n (x + i y) = \frac{1}{2 i} l n (\frac{x + i y - i}{x + i y + i})

l n (\frac{x + i (y - 1)}{x + i (y + 1)}) = a + i b

\frac{x + i (y - 1)}{x + i (y + 1)} = e^{a} e^{i b}

\frac{(x + i (y - 1)) (x + i (y + 1))}{x^{2} + {(y + 1)}^{2}} = e^{a} e^{i b}

\frac{x^{2} + 2 i x y - (y^{2} - 1)}{x^{2} + {(y + 1)}^{2}} = e^{a} e^{i b}

\frac{x^{2} - y^{2} + 1}{x^{2} + {(y + 1)}^{2}} + i \frac{2 x y}{x^{2} + {(y + 1)}^{2}} = e^{a} e^{i b}

t a n (b) = \frac{2 x y}{x^{2} - y^{2} + 1} \Leftrightarrow b = a r c t a n (\frac{2 x y}{x^{2} - y^{2} + 1})

e^{a} = \sqrt{{(\frac{x^{2} - y^{2} + 1}{x^{2} + {(y + 1)}^{2}})}^{2} + {(\frac{2 x y}{x^{2} + {(y + 1)}^{2}})}^{2}}

e^{a} = \frac{\sqrt{x^{4} + y^{4} - 2 x^{2} y^{2} + 2 x^{2} - 2 y^{2} + 1 + 4 x^{2} y^{2}}}{x^{2} + {(y + 1)}^{2}}

a = l n (\frac{\sqrt{{(x^{2} + y^{2} + 1)}^{2} - 4 y^{2}}}{x^{2} + {(y + 1)}^{2}})

t a n (x + i y) = \frac{1}{2 i} (l n (\frac{\sqrt{{(x^{2} + y^{2} + 1)}^{2} - 4 y^{2}}}{x^{2} + {(y + 1)}^{2}}) + i a r c t a n (\frac{2 x y}{x^{2} - y^{2} + 1})

= \frac{1}{2} a r c t a n (\frac{2 x y}{x^{2} - y^{2} + 1}) - i \frac{1}{2} l n (\frac{\sqrt{{(x^{2} + y^{2} + 1)}^{2} - 4 y^{2}}}{x^{2} + {(y + 1)}^{2}})

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