Can we write $z=\sum_{n=1}^\infty a_n\text{trig}_n(b_nz)$ where constants $a_n$ and $b_n$, and trig functions $\text{trig}_n$, are independent of $z$?

Out of curiosity, does anyone happen to know if it's possible to write a complex number $$z\in\mathbb{C}$$ in terms of a series in trig functions ($$\sin$$, $$\cos$$, $$\tan$$, $$\sec$$, $$\csc$$, $$\cot$$, etc.) and/or hyperbolic trig functions ($$\sinh$$, $$\cosh$$, $$\tanh$$, sech, csch, $$\coth$$, etc.)? Ie, something of the form $$z=\sum_{n=1}^\infty a_n\text{trig}_n(b_nz),$$ where each $$\text{trig}_n$$ is a trig or hyperbolic trig function (which doesn't necessarily have to be the same for each $$n$$) and $$a_n$$,$$b_n$$ are (possibly compex) constants (independent of $$z$$)?

• Sounds like Fourier series might be a topic worth looking into since they can express "nice enough" functions as a (possibly infinite) sum of sine and/or cosine functions May 13, 2020 at 21:57
• But I think that really only works for periodic functions, and $z$ is not periodic. May 13, 2020 at 22:06

Choosing a definite argument $$\;t\in[0,2\pi)\;$$ for $$\;z\in\Bbb C\;$$ , we can write
$$z=|z|e^{it}=|z|\cos t+i|z|\sin t$$
There, you wrote $$\;z\in\Bbb C\;$$ as a sum of trigonomeric functions (multiplied each by some constant). Certainly, the case $$\;z=0\;$$ should be dealt with separatedly...
• But I wanted all the $z$ dependence inside the trig functions. The problem is trivial otherwise. May 13, 2020 at 22:08
• It means something of the form $a_n\cdot \text{trig}(b_n z)$ with $a_n$ and $b_n$ (possibly complex) constants. I will edit my question to clarify. May 13, 2020 at 22:20
• ...and $$z=|z|\cos\left(\arg z\right)+i|z|\sin\left(\arg z\right)\ldots?$$ Here, $$a_1=|z|,\,b_1=\arg z\,,\,\,a_2=i|z|\,,\,\,b_2=\arg z$$ and, of course, $\;a_n=b_n=0\;$ for $\;n\ge3\;$ ...This, of course, is exactly the same that in my answer but written in a much more cumbersome way. If you'd add some context to your question then it'd be probably easier to know what are you expecting... May 13, 2020 at 23:08
• But your $a_1$, $a_2$ depend on $z$... I asked for $a_n$ and $b_n$ to be constants, i.e., independent of $z$. May 13, 2020 at 23:16