My Engineering Mathematics teacher have a very novel method of teaching. He thinks that while it's all good and well to learn the analytical side of math, he stresses more on the theory, than the application. Thus he gave us these questions that we must answer without a single equation. It's new and I sure enjoy the concept, but I'm not sure if I got it right. Could someone check my answers for me? Thanks!
1. Why there are n solutions to the $n$th root of a complex number?
Because if the solution is some number w to the power of n (a series of natural numbers) then for each value of w there is one and only one solution value.
2. If a complex function is analytic, the function needs to satisfy the Cauchy-Riemann equation. Outline the proof process of the necessary part of this argument.
If a complex function is to be considered analytic, it must have a derivative at all points in its domain. Thus, if we assume a function has a derivative, then, according to the fundamental theorem of calculus, it can be expressed in terms of a limit. We can then divide the limit into real and imaginary components, since it is on a complex plane. And since the derivative exists, the limit also exists. This means that we can take the limit in any direction. If we approach it in a horizontal direction and in a vertical direction, we get two different equations for the derivative of the function. And since they are the meant to be equal, if we equate the real and imaginary terms of both equations, we get the two Cauchy-Riemann equations, thus showing that for a function to be analytic, it must satisfy the Cauchy-Riemann equations. This is the necessary condition for a function to be considered analytic.
3. Outline the proof process of the sufficient part of the argument in problem 2.
The sufficient part of the argument outlined in problem 13-2 is that the partial derivative of the real part u of the function with respect to x is equal to the partial derivative of the imaginary part v of the function with respect to y. Additionally, the partial derivative of u with respect to y should be equal to the negative partial derivative of v with respect to x. Also, all partial derivatives of the function must be continuous. If we supply these substitutions in the Taylor series representation of the function and rearrange, then we can get the fundamental calculus definition of the derivative of the function with respect to a path z. If we substituted so that all partial derivatives are in respect of x, then we get the derivative of the function in the x-direction. The opposite is true if we substituted so that all partial derivatives are in respect of y. And since all the partial derivatives are continuous, the derivative of the function in both cases must exist. Thus, the function is differentiable, thus proving it’s analytic quality. Note that although the Cauchy-Riemann equations are satisfied, it is not compulsory that the function is analytic until the partial derivatives are continuous.
4. Describe the overall procedure of defining various complex functions: exponential, trigonometric, hyperbolic, logarithm, and general power. Discuss differences and similarities of the complex functions compared to real functions.
From what I can observe, the general method of defining a complex function is to replace the x in the function’s real counterpart with a complex number. This method, armed with Euler’s equation, it becomes a simple matter or algebraic manipulation to get the complex functions. The complex exponential function is very similar to the real exponential function in that its derivative is itself and is even equal to the real function if the complex number has no imaginary part. Complex trigonometric functions, along with the hyperbolic functions, are also similar in that if the complex number has no imaginary part, the complex counterparts work exactly the same. Also, the complex trigonometric functions and the hyperbolic functions are entire, have the same derivative as their real counterparts and even hold the same general formulas as the real counterparts. However, the complex logarithmic and general power functions vary from their real counterparts. This is due to the fact that a complex exponential function has an infinite number of solutions, not allowing us to define a complex logarithmic function as we would a real logarithmic function. Thus, complex logarithmic functions differ in that while positive real values yield the same results, negative numbers do not. The same concept applies to complex general power functions (except that in this case the complex logarithmic function has infinitely many solutions) and thus some exponential laws cannot be carried out with complex power functions.