Power rule valid for any complex exponents?

In summary, the power rule for complex exponents states that for any complex number z and any real number n, (z^n)' = n*z^(n-1). This rule is applicable for all complex numbers as long as the exponent is a real number. It is derived using the same principles as the power rule for real exponents, but cannot be extended to include complex exponents with imaginary parts. There are exceptions to the power rule for complex exponents, such as when the exponent is a negative integer, in which case the quotient rule must be used to find the derivative.
  • #1
lolgarithms
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Is the power rule in calculus true for any complex exponents? i.e. does d/dz z^c = cz^(c-1) true for any c, even when c is something like i or 3i-2?
 
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  • #2
Yes, try to prove it using the definition of the complex derivative.
 

1. What is the power rule for complex exponents?

The power rule for complex exponents states that for any complex number z and any real number n, (z^n)' = n*z^(n-1).

2. Is the power rule applicable for all complex numbers?

Yes, the power rule is valid for all complex numbers, as long as the exponent is a real number.

3. How is the power rule derived for complex exponents?

The power rule for complex exponents is derived using the same principles as the power rule for real exponents, by applying the chain rule and the definition of complex numbers.

4. Can the power rule be extended to include complex exponents with imaginary parts?

No, the power rule is only applicable for complex exponents with real numbers. For complex exponents with imaginary parts, the logarithmic differentiation method can be used to find the derivative.

5. Are there any exceptions to the power rule for complex exponents?

Yes, the power rule may not apply for certain cases where the exponent is a negative integer. In such cases, the derivative must be found using the quotient rule.

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