As the inverse tangent function is a multivalued function, we need a branch cut in the complex plane to resolve its values.
We can mathematically express the tan-1 function as,
tan-1 y = 1 / 2 i [ ln ( 1- i y) – ln ( 1 + i y ) ],here, ‘i’ represents for complex numbers. As we discussed the branch cut provides the range for tan-1 function.
The range for x – axis along positive direction is,
Range = ( - ∏/2 , ∏/2 )
The most common range of tan -1 in other quadrants is in the interval (∏ , 0 ).
There are some common values for the inverse tan function as,
tan-1 ( - i ) = - 1 ∞,
tan-1 ( - i ) = - i ∞,
tan -1 0 = 0,
tan -1 i = i ∞,
The differentiation of the inverse tan function is defined mathematically as:
d ( tan-1 y ) / dy = 1 / ( 1+ y2 ).
Similarly, the Integration is obtained by the reverse process,
∫ tan -1 y dy = y tan -1 y – 1 / 2 lm ( 1 + y2 ) + k,
‘k’ is the integration constant.
In the form of complex numbers,
The inverse tan is,
∅=tan -1 ( y / x ),
Here, ∅is the angle measured in the clockwise direction.