My textbook[1] defines analytic with the following notation:
$$z=x+iy$$ $$w=f(z)=u(x,y) + i v(x,y)$$
$$\textrm{Analytic if:}~~~u_x=v_y ~~\textrm{and}~~ u_y = -v_x$$
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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[1]Kreyszig 10th edition page 625.
There are two types of functions here.
The first one is $f\colon U \subseteq\Bbb C \to \Bbb C$, 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 $f\colon I\subseteq \Bbb R \to \Bbb C$, which we call differentiable if when writing $f = u+iv$, both functions $u,v\colon I \to \Bbb R$ 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's theorem, maximum modulus principle, and other results you'll learn further in this course).