Complex Analysis: Power Functions

[@TanmaY_TalK]

5:19 in this observation where a is constant and z is variable =! 0; it is no more power function it becomes polynomial function

[@cameronspalding9792]

@21:00 a+b must be an integer

[@michaelbaum6796]

Brilliant explanation 👍

[@cameronspalding9792]

Technically this function is only multivalved when considered as a function of a: once you’ve fixed a value of log(a), f is single valued

[@Grecks75]

Even with a fixed value of a (and that is what we're talking about here, f(z) is only a function of z), you still need to specify which branch of the complex log function you're using in _your_ definition of the f(z)=a^z exponential function. After making (and documenting) that decision, your exponential function will be single-valued. But depending on that choice, the f(z)=a^z function will take on different values for a given z. On the other hand, if you don't specify which branch of the log function you're using in your definition, the f(z)=a^z function can be considered a multi-valued function of the single variable z in the same way as the complex log function is considered a multi-valued function of its single argument.

[@cameronspalding9792]

@@Grecks75 I don’t think that’s right. f(z) = a^z = exp(log(a)*z). The only ambiguity is which value of log(a) you choose, f should be single valued because exp is single valued

[@Grecks75]

@@cameronspalding9792 I agree with everything you said. Once you fix a specific value for log(a) (this includes chosing a branch for log), the function becomes single-valued. As long as you only fix the value of a but do not fix a specific branch of log, the resulting function will still be multi-valued because of the ambiguity in the value of log(a). I think we both mean the same thing, but have focused on different aspects of the resulting concatenated function.

[@041-tutulmalakar7]

Integrate x^3exp(-2x)sinx from 0 to infinity using differentiation under integral sign . I find this problem in Advance calculus explore book's gamma function chapter

[@arkadelik]

emperor 👑

[@javierpacheco2895]

amazin