What are the differences between e^x and e^ix both graphically and theoretically?And my second question is what is the precise intuition behind rotation of 2D number and exponential function?

Question

faux0101d · Accepted Answer

Hint:       e          i      x        =  cos  ⁡  x  +  i  sin  ⁡  x. This is called the Euler&#039;s formula and sometimes written as cis x.

Alexander Lewis · Accepted Answer

Multiplication by       e          i      θ       (where   θ is real) corresponds to a rotation of   θ about the origin. This is best checked by using the polar form   z  =  r  ⋅      e          i      θ      . Perhaps a more basic approach is using Euler&#039;s formula      e          i      θ        =  cos  ⁡  (  θ  )  +  i  sin  ⁡  (  θ  )and trigonometry (angle sum formulas) to justify why multiplication in polar form works the way it does. If you want to go even deeper, however, you will probably need to use power series to justify Euler&#039;s formula itself.Now, since  i  =  exp  ⁡      (                  i                  π          2                      )    ,we have that ix is obtained from x by a       90    ∘   degree rotation. The maps       e          i      x       and       e    x   are hence related by this rotation.

What are the differences between e^x and e^ix both graphically and theoretically?And my second question is what is the precise intuition behind rotation of 2D number and exponential function?

Answered question

Answer & Explanation

New Questions in Calculus 1