I am attempting to determine the locus of z for arg ⁡ ( z 2 + 1 ). I do know that since z 2 + 1 = ( z + i ) ( z − i ), I could write arg ⁡ ( z 2 + 1 ) = arg ⁡ ( z + i ) + arg ⁡ ( z − i ) but I don’t really know how to figure out the locus of the points from here. How should I approach this? Is there a relationship with the difference in two arguments (arc of a circle)? Thank you!

Question

I am attempting to determine the locus of z for   arg  ⁡  (      z    2    +  1  ). I do know that since       z    2    +  1  =  (  z  +  i  )  (  z  −  i  ), I could write   arg  ⁡  (      z    2    +  1  )  =  arg  ⁡  (  z  +  i  )  +  arg  ⁡  (  z  −  i  ) but I don’t really know how to figure out the locus of the points from here. How should I approach this? Is there a relationship with the difference in two arguments (arc of a circle)? Thank you!

Gary Salinas · Accepted Answer

Let     z  =  x  +  i  y   with     x  ,  y  ∈      R    , and let     arg  ⁡  (      z    2    +  1  )  =  α  .1. If     α  =  ±  π      /    2  , then         z    2    +  1   must lie on the imaginary axis, so         z    2    +  1  =  i  b   for some     b  ∈      R    , which in cartesian coodinates translates to         x    2    −      y    2    +  2  i  x  y  +  1  =  i  b  , so the locus is part of the hyperbola         x    2    −      y    2    +  1  =  0  .2. If     α  ≠  ±  π      /    2  , then     a  =  tan  ⁡  (  α  )  =  Im  ⁡  (  z  )      /    Re  ⁡  (  z  )   is well defined, and the equation becomes:            2      x      y                      x        2            −              y        2            +      1        =  a      ⟺      a      x    2    −  2  x  y  −  a      y    2    +  a  =  0This is a quadratic equation with discriminant         2    2    −  4  ⋅  a  ⋅  (  −  a  )  &amp;gt;  0  , so the locus is again part of a hyperbola.In both cases, the actual locus is the part of the hyperbola lying in the quadrant determined by     α  .

I am attempting to determine the locus of z for arg ⁡ ( z 2

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