suvadip said:I need to prove
\(\displaystyle sin^{-1}(ix)=2n\pi\pm i log (\sqrt{1+x^2}+x)\)
I can prove \(\displaystyle sin^{-1}(ix)=2n\pi+ i log (\sqrt{1+x^2}+x)\)
How to prove the other part. Please help
The notation sin⁻¹(ix) represents the inverse sine function, which is the angle whose sine is ix. In other words, it is the angle that, when plugged into the sine function, would give ix as the output.
The complex logarithm, denoted as log(z), is defined as the inverse function of the exponential function. In this case, the complex logarithm of (√1+x²+x) is equal to the inverse sine of ix, hence the notation sin⁻¹(ix).
The ± sign represents the two possible solutions for the inverse sine function. Since the sine function is periodic with a period of 2π, there are infinitely many angles that have the same sine value. Therefore, the inverse sine function has two solutions for each input.
The proof of sin⁻¹(ix) involves using the definition of the inverse sine function and the properties of complex numbers, such as Euler's formula and the properties of logarithms. It is a complex and intricate proof that requires a strong understanding of complex analysis.
The inverse sine function, including its complex counterpart sin⁻¹(ix), is used in various fields of mathematics and physics. It has applications in trigonometry, calculus, differential equations, and signal processing, among others. It is also an important tool in solving complex equations and understanding the nature of complex numbers.