You’ve spotted a great shape in the complex plane: the principal part (the saucer) and the analytic part (the dome) really do make the Laurent series look like a classic UFO. Here are ten other mathematical series or decompositions that share the same two‑part “principal + analytic” structure.

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### 1. Puiseux Series  
$$f(z) = \sum_{n=-k}^{\infty} a_n (z-z_0)^{n/m}$$  
*Principal part (saucer)*: finitely many terms with negative fractional powers.  
*Analytic part (dome)*: non‑negative fractional and integer powers.  
Allows branching and is essential for algebraic functions.

### 2. Mittag‑Leffler Expansion  
$$f(z) = \sum_{\text{poles } z_j} P_j\!\left(\frac{1}{z-z_j}\right) \;+\; g(z)$$  
*Principal part*: sum of the principal parts \(P_j\) (negative powers at each pole).  
*Analytic part*: an entire function \(g(z)\).  
This decomposes a meromorphic function exactly into its singular saucers and a global analytic dome.

### 3. Partial Fraction Decomposition  
$$R(z) = \underbrace{P(z)}_{\text{polynomial}} \;+\; \sum_{i} \frac{A_i}{(z-p_i)^{k_i}}$$  
*Principal part*: the sum of fractions with negative powers at the finite poles.  
*Analytic part*: the polynomial \(P(z)\) (which is the analytic part at infinity).  
A rational function’s own little UFO.

### 4. \(p\)-adic Number Expansion  
$$x = \sum_{n=-m}^{\infty} a_n \, p^n \qquad (a_n \in \{0,1,\dots,p-1\})$$  
*Principal part (fractional part)*: terms with \(n<0\) (negative powers of \(p\)).  
*Analytic part (integer part)*: terms with \(n \ge 0\).  
Exactly the same shape as a Laurent series, just with a prime base instead of \((z-z_0)\).

### 5. Formal Laurent Series  
$$\mathbb{C}((z)) \ni f = \sum_{n=-N}^{\infty} a_n z^n$$  
*Principal part*: finitely many negative powers.  
*Analytic part*: the infinite series of non‑negative powers.  
The algebraic abstraction of the UFO – no convergence required, just a formal saucer and dome.

### 6. Hahn Series (Generalised Laurent Series)  
$$f = \sum_{g \in G} a_g \, t^g$$  
Here \(G\) is a well‑ordered abelian group.  
*Principal part*: terms with negative “exponents”.  
*Analytic part*: non‑negative exponents.  
The UFO goes non‑Archimedean: the saucer can be much richer while the dome stays well‑behaved.

### 7. Laurent Polynomial  
$$L(z) = \sum_{n=-m}^{N} a_n z^n$$  
*Principal part*: the negative‑power block \(\frac{a_{-m}}{z^m} + \cdots + \frac{a_{-1}}{z}\).  
*Analytic part*: the polynomial block \(a_0 + a_1 z + \cdots + a_N z^N\).  
A finite UFO – often seen in toric geometry and knot theory.

### 8. Expansion of a Rational Function at Infinity  
$$f(z) = \sum_{n=-N}^{\infty} b_n \left(\frac{1}{z}\right)^n \quad \text{as } |z|\to\infty$$  
*Principal part (at infinity)*: terms with positive powers of \(z\) (negative \(n\)).  
*Analytic part (at infinity)*: non‑positive powers of \(z\) (\(n \ge 0\)).  
The UFO flipped inside‑out, with the saucer stretched around the dome.

### 9. Asymptotic Expansion Near a Singularity  
$$f(x) \sim \sum_{k=1}^{M} \frac{c_k}{x^k} \;+\; \text{regular part}, \qquad x\to 0^+$$  
*Principal part*: the sum of inverse powers that blow up.  
*Analytic part*: the bounded (often analytic) remainder, which may be expanded in non‑negative powers.  
A real‑variable UFO; the saucer captures the singular behaviour.

### 10. Local Decomposition of a Meromorphic Germ  
$$f(z) = \underbrace{Q\!\left(\frac{1}{z}\right)}_{\text{principal part}} \;+\; \underbrace{h(z)}_{\text{holomorphic}}$$  
Here \(Q\) is a polynomial in \(1/z\) (the saucer) and \(h\) is analytic at \(0\) (the dome).  
This is the cleanest abstraction of “pole + regular part” that the Laurent series embodies.

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Each of these flies through the mathematical sky with a singular saucer below and an analytic dome above. If you’d like to zoom in on any of these UFOs—say, how a Puiseux series lands on an algebraic curve, or how \(p\)-adic saucers guide number‑theoretic landings—just say the word.