Question

tan h(x+iy) seperate real and imaginary ​

Answer 1

Answer:

Let α+iβ=tan

−1

(x+iy)

Then α−iβ=tan

−1

(x−iy)

Adding, we get

2α=tan

−1

(x+iy)+tan

−1

(x−iy)

⇒tan

−1

(

1−(x+iy)(x−iy)

x+iy+x−iy

)

∴ Real part α=

2

1

tan

−1

(

1−(x+iy)(x−iy)

x+iy+x−iy

)

Subtracting, 2iβ=tan

−1

(x+iy)−tan

−1

(x−iy)

⇒2iβ=tan

−1

(

1+(x+iy)(x−iy)

(x+iy)−(x−iy)

)

⇒2iβ=tan

−1

(

1+x

2

+y

2

2iy

)

⇒2iβ=itanh

−1

(

1+x

2

+y

2

2y

)

∴ Imaginary part β=

2

1

tanh

−1

(

1+x

2

+y

2

2y