Entire Slice Regular Functions pp 7-30 | Cite as

# Slice Regular Functions: Algebra

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## Abstract

This chapter contains the definition and some basic results on slice regular functions. In particular, we discuss the subclass of slice regular intrinsic functions which possess additional properties. This subclass contains the transcendental functions like the quaternionic exponential, sine and cosine. We also introduce a suitable multiplication and a notion of composition which preserve slice regularity.

## Keywords

Power Series Regular Function Formal Power Series Clifford Algebra Representation Formula## 2.1 Definition and Main Results

In this chapter, we will present some basic material on slice regular functions, a generalization of holomorphic functions to the quaternions.

*i*,

*j*,

*k*satisfy

*I*. Any nonreal quaternion \(q=x_0 + i x_{1} +j x_{2} + k x_{3}\) is uniquely associated to the element \(I_q\in \mathbb {S}\) defined by

*q*belongs to the complex plane \(\mathbb {C}_{I_q}\).

### Definition 2.1

*U*be an open set in \(\mathbb {H}\) and let \(f:\, U\rightarrow \mathbb {H}\) be real differentiable. The function

*f*is said to be (left) slice regular or (left) slice hyperholomorphic if for every \(I\in \mathbb {S}\), its restriction \(f_I\) to the complex plane \({\mathbb {C}}_{I}=\mathbb {R}+ I \mathbb {R}\) passing through origin and containing

*I*and 1 satisfies

*U*will be denoted by \(\mathscr {R}(U)\).

*U*if

It is immediate to verify that:

### Proposition 2.1

Let *U* be an open set in \(\mathbb H\). Then \(\mathscr {R}(U)\) is a right linear space on \(\mathbb H\).

*I*-derivative of

*f*at a point \(q=x+I y\) is defined by

*I*-derivative of

*f*at \(q=x+I y\) is defined by

### Definition 2.2

*U*be an open set in \(\mathbb {H}\), and let \(f:U \rightarrow \mathbb {H}\) be a slice regular function. The slice derivative \(\partial _s f\) of

*f*, is defined by:

It is important to note that if *f*(*q*) is a slice regular function then also \(f'(q)\) is a slice regular function.

Let \(I,J\in \mathbb {S}\) be such that *I* and *J* are orthogonal, so that \(I,J,IJ=K\) is a basis of \(\mathbb H\) and write the restriction \(f_I(x+Iy)=f(x+Iy)\) of *f* to the complex plane \(\mathbb C_I\) as \(f=f_0+If_1+Jf_2+Kf_3\), where \(f_0,\ldots ,f_3\) are \(\mathbb R\)-valued. In alternative, it can also be written as \(f=F+GJ\) where \(f_0+If_1=F\), and \(f_2+If_3=G\) are \(\mathbb C_I\)-valued. This observation immediately gives the following result:

### Lemma 2.1

**(Splitting Lemma)**If

*f*is a slice regular function on

*U*, then for every \(I \in \mathbb {S}\), and every \(J\in \mathbb {S}\), perpendicular to

*I*, there are two holomorphic functions \(F,G:U\cap \mathbb {C}_I \rightarrow \mathbb {C}_I\) such that for any \(z=x+Iy\)

### Proof

*f*is slice regular, we know that

We now consider slice regular functions on open balls *B*(0; *r*) centered at the origin with radius \(r>0\). We have:

### Theorem 2.1

*B*(0;

*r*). Moreover, \(f\in \mathscr {C}^\infty (B(0;R))\).

### Proof

*F*,

*G*are holomorphic. Then

*F*and

*G*admit power series expansions which converge uniformly and absolutely on any compact set in \(B(0;r)\cap \mathbb C_I\):

*f*is infinitely differentiable in

*B*(0;

*r*) by the uniform convergence on compact subsets. \(\square \)

The proof of the following result is the same as the proof in the complex case.

### Theorem 2.2

*B*(0;

*r*). Its sum defines a slice regular function on

*B*(0;

*r*).

### Definition 2.3

A function slice regular on \(\mathbb H\) will be called entire slice regular or entire slice hyperholomorphic.

Every entire regular function admits power series expansion of the form (2.1) which converges everywhere in \(\mathbb H\) and uniformly on the compact subsets of \(\mathbb H\).

## 2.2 The Representation Formula

Slice regular functions possess good properties on specific open sets that are called axially symmetric slice domains. On these domains, slice regular functions satisfy the so-called Representation Formula which allows to reconstruct the values of the function once that we know its values on a complex plane \(\mathbb C_I\). As we shall see, this will allow also to define a suitable notion of multiplication between two slice regular functions.

### Definition 2.4

Let \(U \subseteq \mathbb H\). We say that *U* is axially symmetric if, for every \(x+Iy \in U\), all the elements \(x+Jy\) for \(J\in \mathbb {S}\) are contained in *U*. We say that *U* is a *slice domain* if it is a connected set whose intersection with every complex plane \(\mathbb C_I\) is connected.

### Definition 2.5

Given a quaternion \(q=x+Iy\), the set of all the elements of the form \(x+Jy\) where *J* varies in the sphere \(\mathbb {S}\) is a two-dimensional sphere denoted by \([x+Iy]\) or by \(x+\mathbb {S}y\).

The Splitting Lemma allows us to prove:

### Theorem 2.3

**(Identity Principle).** Let \(f:U\rightarrow \mathbb {H}\) be a slice regular function on a slice domain *U*. Denote by \(Z_f=\{q\in U : f(q)=0\}\) the zero set of *f*. If there exists \(I \in \mathbb {S}\) such that \(\mathbb {C}_I \cap Z_f\) has an accumulation point, then \(f\equiv 0\) on *U*.

### Proof

*f*to \(U\cap \mathbb C_I\) is such that \(F,G:\, U\cap \mathbb C_I \rightarrow \mathbb C_I\) are holomorphic functions. Under the hypotheses,

*F*and

*G*vanish on a set \(\mathbb {C}_I \cap Z_f\) which has an accumulation point so

*F*and

*G*are both identically zero. So \(f_I\) vanishes on \(U\cap \mathbb R\) and, in particular,

*f*vanishes on \(U\cap \mathbb R\). Thus the restriction of

*f*to any other complex plane \(\mathbb C_L\) vanishes on a set with an accumulation point and so \(f_L\equiv 0\). Since

*f*vanishes on

*U*. \(\square \)

The following result shows that the values of a slice regular function defined on an axially symmetric slice domain can be computed from the values of its restriction to a complex plane:

### Theorem 2.4

**(Representation Formula).**Let

*f*be a slice regular function defined on an axially symmetric slice domain \(U\subseteq \mathbb {H}\). Let \(J\in \mathbb {S}\) and let \(x\pm Jy\in U\cap \mathbb C_J\). Then the following equality holds for all \(q=x+Iy \in U\):

*I*, such for any \(K \in \mathbb {S}\) we have

### Proof

*q*is real and the proof is immediate. Otherwise let us define the function \(\psi :U \rightarrow \mathbb {H}\) by

*U*, the first part of the assertion follows from the Identity Principle. Since

*f*is regular on

*U*, for any \(I\in \mathbb {S}\) we have, on \(U\cap \mathbb C_{I}\),

Some immediate consequences are the following corollaries:

### Corollary 2.1

A slice regular function \(f : U\rightarrow \mathbb {H}\) on an axially symmetric slice domain is infinitely differentiable on *U*. It is also real analytic on *U*.

### Corollary 2.2

*f*is affine in \(I\in \mathbb {S}\) on each 2-sphere \([x_0+Iy_0]\) and the image of the \(2-\)sphere \([x_0+Iy_0]\) is the set \([a+Ib]\).

### Corollary 2.3

Let \(U\subseteq \mathbb {H}\) be an axially symmetric slice domain and let \(f : U \rightarrow \mathbb {H}\) be a slice regular function. If \(f(x+Jy)=f(x+Ky)\) for \(J\ne K\) in \(\mathbb {S}\), then *f* is constant on \([x+Iy]\). In particular, if \(f(x+Jy)=f(x+Ky)=0\) for \(J\ne K\) in \(\mathbb {S}\), then *f* vanishes on the whole \(2-\)sphere \([x+Iy]\).

### Corollary 2.4

*f*is slice regular if and only if there exist two differentiable functions \(\alpha , \beta : D\subseteq \mathbb {R}^2 \rightarrow \mathbb {H}\) satisfying \(\alpha (x,y)=\alpha (x,-y)\), \(\beta (x,y)=-\beta (x,-y)\) and the Cauchy–Riemann system

### Proof

A function of the form (2.7) where \(\alpha \) and \(\beta \) satisfy the hypothesis in the statement is clearly slice regular. Conversely, a slice regular function on an axially symmetric slice domain satisfies the Representation formula and thus it is of the form (2.7), where \(\alpha \) and \(\beta \) satisfy the Cauchy–Riemann system. The conditions \(\alpha (x,y)=\alpha (x,-y)\), \(\beta (x,y)=-\beta (x,-y)\) can be easily verified from the definition of \(\alpha \) and \(\beta \) given in the Representation Formula. \(\square \)

### Remark 2.1

*I*and we write both \(\alpha \) and \(\beta \) into their real components (omitting, for simplicity, the arguments of the functions):

*x*,

*y*.

The previous remark implies a stronger version of the Splitting Lemma for slice regular functions. To prove the result, we need to recall that complex functions defined on open sets \(G\subset \mathbb {C}\) symmetric with respect to the real axis and such that \(\overline{f(\bar{z})}=f(z)\) are called in the literature intrinsic, see [143].

### Proposition 2.2

**(Refined Splitting Lemma).**Let

*U*be an axially symmetric slice domain in \(\mathbb {H}\) and let \(f\in \mathscr {R}(U)\). For any \(I, J\in \mathbb {S}\) with

*J*orthogonal to

*I*, there exist four holomorphic intrinsic functions \(h_\ell :\ U\cap \mathbb {C}_I\rightarrow \mathbb {C}_I\), \(\ell =0,\ldots , 3\) such that

### Proof

*y*implies that \(a_\ell (x,y)\) and \(b_\ell (x,y)\) are even and odd, respectively, in the variable

*y*. Thus

Another important consequence of the Representation Formula is the following result, which allows to obtain the slice regular extension of holomorphic maps:

### Corollary 2.5

*U*be the axially symmetric slice domain defined by

*f*to

*U*.

### Definition 2.6

*U*is the axially symmetric completion of \(U_J\) in \(\mathbb H\).

Corollary 2.4 implies that slice regular functions are a subclass of the following set of functions:

### Definition 2.7

Let \(U\subseteq \mathbb H\) be an axially symmetric open set. Functions of the form \(f(q)=f(x+Iy)=\alpha (x,y)+I\beta (x,y)\), where \(\alpha \) \(\beta \) are \(\mathbb H\)-valued functions such that \(\alpha (x,y)=\alpha (x,-y)\), \(\beta (x,y)=-\beta (x,-y)\) for all \(x+Iy\in U\) are called *slice functions*. If \(\alpha \) and \(\beta \) are continuous then *f* is said to be a slice continuous function.

### Theorem 2.5

**(General Representation Formula).**Let \(U \subseteq \mathbb {H}\) be an axially symmetric slice domain and let \(f\in \mathscr {R}(U)\). The following equality holds for all \(q=x+I y \in U\), \(J,K\in \mathbb S\):

### Proof

*q*is real the proof is immediate. Otherwise, for all \(q=x+yI\), we define the function

*U*, the first part of the assertion will follow from the Identity Principle for slice regular functions. Indeed, since

*f*is slice regular on

*U*, for any \(L\in \mathbb {S}\) we have \(\frac{\partial }{\partial x}f(x+Ly)= -L\frac{\partial }{\partial y}f(x+Ly)\) on \(U\cap \mathbb {C}_{L}\); hence

## 2.3 Multiplication of Slice Regular Functions

*i*being an imaginary unit) and the pointwise product

*iq*, which is not slice regular.

In the case of slice regular functions defined on axially symmetric slice domains, we can define a suitable product, called \(\star \)-product, which preserves slice regularity. This product extends the very well known product for polynomials and series with coefficients in a ring, see e.g., [92] and [133].

*U*defined, according to the extension formula (2.10) by

*U*.

### Definition 2.8

*f*and

*g*. This product is also called \(\star \)-product.

### Remark 2.2

It is immediate to verify that the \(\star \)-product is associative, distributive but, in general, not commutative.

### Remark 2.3

Let *H*(*z*) be a \(\mathbb C_I\)-valued holomorphic function in the variable \(z\in \mathbb {C}_I\) and let \(J\in \mathbb {S}\) be orthogonal to *I*. Then by the definition of \(\star \)-product we obtain \(J\star H(z)=\overline{H(\bar{z})}J\).

To define the inverse of a function with respect to the \(\star \)-product we need to introduce the so-called conjugate and symmetrization of a slice regular function. These two notions, as we shall see, will be important also for other purposes.

Using the notations above we have:

### Definition 2.9

*f*as

### Remark 2.4

*f*in the form \(f(x+Iy)=\alpha (x,y)+I\beta (x,y)\), then it is possible to show that

### Remark 2.5

*p*is a fixed quaternion and \(I\in \mathbb S\) there exists \(J\in \mathbb S\) such that \(Ip=pJ\). Thus we have

Using the notion of \(\star \)-multiplication of slice regular functions, it is possible to associate to any slice regular function *f* its “symmetrization” also called “normal form”, denoted by \(f^s\). We will show that all the zeros of \(f^s\) are spheres of type \([x+Iy]\) (real points, in particular) and that, if \(x+Iy\) is a zero of *f* (isolated or not) then \([x+Iy]\) is a zero of \(f^s\).

### Definition 2.10

*f*.

### Remark 2.6

Note that formula (2.18) yields that, for all \(I\in \mathbb {S}\), \(f^s( U\cap \mathbb {C}_I)\subseteq \mathbb {C}_I\).

We now show how the conjugation and the symmetrization of a slice regular function behave with respect to the \(\star \)-product:

### Proposition 2.3

### Proof

### Proposition 2.4

Let \(U\subseteq \mathbb {H}\) be an axially symmetric slice domain and let \(f : U \rightarrow \mathbb {H}\) be a slice regular function. The function \((f^s)^{-1}\) is slice regular on \(U\setminus \{ q\in \mathbb {H}\ : \ f^s(q)=0\}\).

### Proof

The function \(f^s\) is such that \(f^s( U\cap \mathbb {C}_I)\subseteq \mathbb {C}_I\) for all \(I\in \mathbb {S}\) by Remark 2.6. Thus, for any given \(I\in \mathbb {S}\), the Splitting Lemma implies the existence of a holomorphic function \(F:\ U\cap \mathbb {C}_I\rightarrow \mathbb {C}_I\) such that \(f^s_I(z)=F(z)\) for all \(z\in U\cap \mathbb {C}_I\). The inverse of the function *F* is holomorphic on \( U\cap \mathbb {C}_I\) outside the zero set of *F*. The conclusion follows by the equality \((f_I^s)^{-1}=F^{-1}\). \(\square \)

The \(\star \)-product can be related to the pointwise product as described in the following result:

### Theorem 2.6

### Proof

*g*, we get

*I*is arbitrary proves the assertion.

An immediate consequence is the following:

### Corollary 2.6

If \((f\star g)(q)=0\) then we have that either \(f(q)=0\) or \(f(q)\not =0\) and \(g(f(q)^{-1}qf(q))=0\).

## 2.4 Quaternionic Intrinsic Functions

*U*, denoted by \(\mathscr {N}(U)\), is defined as follows:

### Remark 2.7

If *U* is axially symmetric and if we denote by \(\overline{q}\) the conjugate of a quaternion *q*, it can be shown that a function *f* belongs to \(\mathscr {N}(U)\) if and only if it satisfies \(f(q)=\overline{f(\bar{q})}\). In analogy with the complex case, we say that the functions such that \(f(q)=\overline{f(\bar{q})}\) are *quaternionic intrinsic*.

If one considers a ball *B*(0; *R*) with center at the origin, it is immediate that a function slice regular on the ball belongs to \(\mathscr {N}(B(0;R))\) if and only if its power series expansion has real coefficients. Such functions are also said to be *real*. More in general, if *U* is an axially symmetric slice domain, then \(f\in \mathscr {N}(U)\) if and only if \(f(q)=f(x+Iy)=\alpha (x,y)+I\beta (x,y)\) with \(\alpha \), \(\beta \) real valued, in fact we have:

### Proposition 2.5

### Proof

If \(\alpha \), \(\beta \) are real valued and satisfy the Cauchy–Riemann system, then trivially *f* defined by (2.21) belongs to \(\mathscr {N}(U)\). Conversely, assume that the slice function *f* belongs to \(\mathscr {N}(U)\). Then \(f_I(z)=F(z)=\alpha (x,y)+I\beta (x,y)\) satisfies the Cauchy–Riemann equation and so the pair \(\alpha \), \(\beta \) satisfies the Cauchy–Riemann system. By hypothesis, for any fixed \(I\in \mathbb S\) the restriction \(f_I\) takes \(\mathbb C_I\) to itself, and so the functions \(\alpha \), \(\beta \) are \(\mathbb C_I\)-valued. By the arbitrariness of *I* it follows that \(\alpha \) and \(\beta \) are real valued. \(\square \)

### Remark 2.8

Let us now recall the definition of quaternionic logarithm, see [109], [127], which is another example of a quaternionic intrinsic function. It is the inverse function of the exponential function \(\exp (q)\) in \(\mathbb {H}\).

### Definition 2.11

*logarithm*) on

*U*a function \(f:U \rightarrow \mathbb {H}\) such that for every \(q\in U \)

### Definition 2.12

The function \(\arccos (\mathrm{Re}(q)/|q|)\) will be called the principal quaternionic argument of *q* and it will be denoted by \(\arg _{\mathbb {H}}(q)\) for every \(q\in \mathbb {H}\setminus \{0\}\).

Below we define the principal quaternionic logarithm.

### Definition 2.13

*q*as

### Remark 2.9

It is easy to directly verify that the principal quaternionic logarithm coincides with the principal complex logarithm on the complex plane \(\mathbb C_I\), for any \(I\in \mathbb S\).

We now go back to the properties of intrinsic slice regular functions. The following result has an elementary proof which is left to the reader.

### Proposition 2.6

**(Algebraic properties).** Let *U*, \(U'\) be two open sets in \(\mathbb H\).

- (1)
Let

*f*and \(g\in \mathscr {N}(U)\), then \(f+g\in \mathscr {N}(U)\) and \(fg=gf\in \mathscr {N}(U)\). - (2)
Let

*f*and \(g\in \mathscr {N}(U)\) such that \(g(q)\not =0\) for all \(q\in U\), then \(g^{-1}f=fg^{-1}\in \mathscr {N}(U)\). - (3)
Let \(f\in \mathscr {R}(U')\), \(g\in \mathscr {N}(U)\) with \(g(U)\subseteq U'\). Then

*f*(*g*(*q*)) is slice regular for \(q\in U\).

### Proposition 2.7

### Proof

Similarly one can see that all the other intersections between \(\mathscr {N}(U)\), \(\mathscr {N}(U)I\), \(\mathscr {N}(U)J\), \(\mathscr {N}(U)IJ\), are reduced to the zero vector and the statement follows. \(\square \)

## 2.5 Composition of Power Series

In this section, we introduce and study a notion of composition of slice regular functions, see also [93]. As it is well known, the composition \(f\circ g\) of two slice regular functions is not, in general, slice regular, unless *g* belongs to the subclass of quaternionic intrinsic functions (see Proposition 2.6). Our choice of the notion of composition is based on the fact that slice regularity is not preserved by the pointwise product, but is preserved by the \(\star \)-product. Thus the power of a function is slice regular only if it is computed with respect to the \(\star \)-product and we will write \((w(q))^{\star n}\) to denote that we are taking the *n*-th power with respect to this product. To introduce the notion of composition, we first treat the case of formal power series.

### Definition 2.14

### Remark 2.10

When \(w \in {\mathscr {N}}(B(0;1))\), then \(g\bullet w=g\circ w\) where \(\circ \) represents the standard composition of two functions. Moreover, when \(w \in {\mathscr {N}}(B(0;1))\) then \((w(q))^{\star n}=(w(q))^n\) so, in particular, \(q^{\star n}=q^n\).

### Remark 2.11

*order*of a series \(f(q)=\sum _{n=0}^{\infty } q^{n}a_{n}\) can be defined as in [37] and we denote it by \(\omega (f)\). It is the lowest integer

*n*such that \(a_n\not =0\) (with the convention that the order of the series identically equal to zero is \(+\infty \)). Assume to have a family \(\{f_i\}_{i\in \mathscr {I}}\) of power series where \(\mathscr {I}\) is a set of indices. This family is said to be

*summable*if for any \(k\in \mathbb N\), \(\omega (f_i)\ge k\) for all except a finite number of indices

*i*. By definition, the sum of \(\{f_i\}\) where \(f_i(q)=\sum _{n=0}^{\infty } q^{n}a_{i,n}\) is

*n*just a finite number of \(a_{i,n}\) are nonzero.

### Remark 2.12

In Definition 2.14, we require the hypothesis \(b_0=0\). This is necessary in order to guarantee that the minimum power of *q* in the term \((w(q))^{\star n}\) is at least \(q^n\) or, in other words, that \(\omega (w(q)^{\star n})\ge n\) for all indices. With this hypothesis, the series \(\sum _{n=0}^\infty (w(q))^{\star n}a_n\) is summable according to Remark 2.11, and we can regroup the powers of *q*.

### Proposition 2.8

The composition \(\bullet \) is, in general, not associative.

However, we will prove that the composition is associative in some cases and to this end we need a preliminary lemma.

### Lemma 2.2

- (1)
\((f_1+f_2)\bullet g= f_1\bullet g+f_2\bullet g\);

- (2)
if

*g*has real coefficients \((f_1\star f_2)\bullet g=(f_1 \bullet g)\star (f_2 \bullet g)\); - (3)
if

*g*has real coefficients \(f^{\star n}\bullet g=(f \bullet g)^{\star n}\). - (4)
if \(\{f_i\}_{i\in \mathscr {I}}\) is a summable family of power series then \(\{f_i\bullet g\}_{i\in \mathscr {I}}\) is summable and \((\sum _{i\in \mathscr {I}} f_i) \bullet g=\sum _{i\in \mathscr {I}} (f_i\bullet g).\)

### Proof

*g*are real we have:

*n*-th power. Let us show that it holds for \((n+1)\)-th power. Let us compute

*q*involves just a finite number of the coefficients \(a_{i,n}\) so we can apply the associativity of the addition in \(\mathbb H\) and so (2.22) and (2.23) are equal. \(\square \)

### Proposition 2.9

If \(f(q)=\sum _{n=0}^{\infty } q^{n}a_{n}\), \(g(q)=\sum _{n=1}^{\infty } q^{n}b_{n}\), \(w(q)=\sum _{n=1}^{\infty }q^{n} c_{n}\) and *w* has real coefficients, then \( (f\bullet g) \bullet w = f\bullet (g \bullet w). \)

### Proof

*f*as the sum of the summable family \(\{q^na_n\}\) and using the first part of the proof:

The discussion, so far, was on formal power series without specifying the set of convergence. We now consider this aspect, by proving the following result which is classical for power series with coefficients in a commutative ring, see [37].

### Proposition 2.10

Let \(f(q)= \sum _{n=0}^\infty q^n a_n\) and \(g(q)= \sum _{n=1}^\infty q^n b_n\) be convergent in the balls of nonzero radius *R* and \(\rho \), respectively, and let \(h(q)=(f\bullet g)(q)\). Then the radius of convergence of *h* is nonzero and it is equal to \(\sup \{r> 0 \ : \ \sum _{n=1}^{\infty } r^{n}|b_{n}|<R \}\).

### Proof

*g*is converging on a ball of finite radius, there exists a positive number

*r*, sufficiently small, such that \(\sum _{n=1}^\infty r^n |b_n| \) is finite. Moreover, \(\sum _{n=1}^\infty r^n |b_n|=r\sum _{n=1}^\infty r^{n-1} |b_n|\rightarrow 0\) for \(r\rightarrow 0\) and so there exists

*r*such that \(\sum _{n=1}^\infty r^n |b_n| <R\). Thus, from (2.25), we deduce

*r*. \(\square \)

Using this notion of composition, it is also possible to define, under suitable conditions, left and right inverses of a power series:

### Proposition 2.11

Let \(g:\, B(0;R)\rightarrow \mathbb {H}\), \(R>0\), be a function slice regular of the form \(g(q)=\sum _{n=0}^\infty q^n a_n\).

- (1)
There exists a power series \(g_r^{-\bullet }(q)=\sum _{n=0}^\infty q^n b_n\) convergent in a disc with positive radius, such that \((g\bullet g_r^{-\bullet })(q)=q\) and \(g_r^{-\bullet }(0)=0\) if and only if \(g(0)=0\) and \(g'(0)\not =0\).

- (2)
There exists a power series \(g_l^{-\bullet }(q)=\sum _{n=0}^\infty q^n b_n\) convergent in a disc with positive radius, such that \((g_l^{-\bullet }\bullet g)(q)=q\) and \(g_l^{-\bullet }(0)=0\) if and only if \(g(0)=0\) and \(g'(0)\not =0\).

### Proof

*g*by construction.

*g*as follows: set

Point (2) can be proved with similar computations and the function \(g_l^{-\bullet }\) so obtained is a left inverse of *g*. \(\square \)

### Remark 2.13

*f*(

*q*) be another entire function, for example a polynomial. It is then possible to define the composite functions

**Comments to Chapter 2**. The material in this chapter comes from several papers. The theory of slice regular functions started in [110] and [111] for functions defined on balls centered at the origin and then evolved in a series of papers, among which we mention [46], [47], [58], into a theory on axially symmetric domains. The theory was developed also for functions with values in a Clifford algebra, see [72], [73], and for functions with values in a real alternative algebra, see [118]. The composition of slice regular functions which appears in this chapter is taken from [93].