examples of Cauchy-Riemann equations
We now separate real from imaginary parts. Letting and be real variables we have
Since and . the Cauchy-Riemann relations are seen to be satisfied.
Next, consider the complex conjugation function. In this case, , so we have and . Taking derivatives,
Because , the Cauchy-Riemann equations are not satisfied do the conjugation function is not holomorphic. Likewise, one can show that the functions which appear in complex analysis are not holomorphic.
For our next example, we try a polynomial. Let . Writing and , we find that and . Taking partial derivatives, one can confirm that the Cauchy-Riemann equations are satisfied, so we have a holomorphic function.
More generally, we can show that all complex polynomials are holomorphic. Since the Cauchy-Riemann equations are linear, it suffices to check that integer powers are holomorphic. We can do this by an induction argument. That satisfies the equations is trivial and we have shown that also satisfies them. Let us assume that happens to satisfy the Cauchy-Riemann equations for a particular value of and write
By elementary algebra, we have
By elementary calculus, we have
Since the terms in parentheses are zero on account of and satisfying the Cauchy-Riemann equations, it follows that and also satisfy the Cauchy-Riemann equations. By induction, is holomorphic for all positive integers .
As our next example, we consider the complex square root. As shown in the entry taking square root algebraically, we have the following equality:
Differentiating and simplifying,
Pulling out a common factor and placing over a common denominator,
so the Cauchy-Riemann equations are satisfied. More generally, it can be shown that all complex algebraic functions and fractional powers satisfy the Cauchy-Riemann equations. However, as suggested by the above derivation, a direct verification could be tedious, so it is better to use an indirect approach.
Thus we see that the complex exponential function is holomorphic.
The complex logarithm may be defined as . Hence we have
Hence the complex logarithm is holomorphic.
|Title||examples of Cauchy-Riemann equations|
|Date of creation||2013-03-22 17:38:05|
|Last modified on||2013-03-22 17:38:05|
|Last modified by||rspuzio (6075)|