Determine if a complex parametric function is analytic. z = x + i y w = f ( z ) = u ( x , y ) + i v ( x , y ) Analytic if: u x = v y and u y = − v x This is not to hard for most problems. However, what if z is in parametric form? For example, z ( t ) = cos ⁡ ( t ) + i sin ⁡ ( t )Is it simply analytic if u t = v t ?

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Determine if a complex parametric function is analytic.  z  =  x  +  i  y  w  =  f  (  z  )  =  u  (  x  ,  y  )  +  i  v  (  x  ,  y  )      Analytic if:                 u    x    =      v    y              and              u    y    =  −      v    x  This is not to hard for most problems. However, what if z is in parametric form? For example,  z  (  t  )  =  cos  ⁡  (  t  )  +  i  sin  ⁡  (  t  )Is it simply analytic if       u    t    =      v    t   ?

Eleanor Luna · Accepted Answer

Step 1There are two types of functions here.The first one is   f  :  U  ⊆      C    →      C   , and you say that f is complex-analytic if the Cauchy-Riemann equations hold in the open set U, for example.The second one is   f  :  I  ⊆      R    →      C   , which we call differentiable if when writing   f  =  u  +  i  v , both functions   u  ,  v  :  I  →      R   are differentiable in the sense you learnt in single-variable calculus. There is no concept of holomorphy or complex-analiticity in this case. However, you can say that this f is real-analytic if both u and v are real-analytic.The two notions of analiticity are distinct, with the complex version being much more rigid (think of Liouville&#039;s theorem, maximum modulus principle, and other results you&#039;ll learn further in this course).

Determine if a complex parametric function is analytic. z = x + i y w

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