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Sign function

The sign function, also known as the signum function and denoted \sgn(x), is a fundamental piecewise-defined function in mathematics that determines the sign of a real number x, returning $1 if x > 0, -1 if x < 0, and $0 if x = 0. This function can also be expressed as \sgn(x) = \frac{x}{|x|} for x \neq 0, with the value at zero specified separately to ensure completeness. It serves as a basic building block in real analysis, providing a simple way to categorize numbers based on their polarity relative to zero. Key properties of the signum function include its odd symmetry, where \sgn(-x) = -\sgn(x) for all real x, reflecting its antisymmetric behavior around the origin. The function is discontinuous at x = 0, as the left-hand limit \lim_{x \to 0^-} \sgn(x) = -1 and the right-hand limit \lim_{x \to 0^+} \sgn(x) = 1 differ, while \sgn(0) = 0. Additionally, it relates directly to other elementary functions, such as the absolute value |x| = x \cdot \sgn(x) and the Heaviside step function, often defined as H(x) = \frac{1 + \sgn(x)}{2} for x \neq 0 (yielding 0 for x < 0 and 1 for x > 0), with H(0) = 1/2 or 1 depending on convention. These relations highlight its role in constructing more complex expressions in and beyond. The finds extensive applications across and , including in the definition and integration of piecewise continuous functions, where it helps handle discontinuities in integrands. In and communications, it is used to extract the of signals, facilitating tasks like and phase detection. It also appears in control systems for modeling switching behaviors and in , where the in the involves multiplication by -i \sgn(\omega). Furthermore, approximations of \sgn(x) by polynomials or entire functions are studied for numerical methods and optimization problems.

Definition

Real-Valued Case

The sign function for real numbers, commonly denoted as \operatorname{sgn}(x), is defined piecewise as follows: \operatorname{sgn}(x) = 1 if x > 0, \operatorname{sgn}(x) = -1 if x < 0, and \operatorname{sgn}(x) = 0 if x = 0. This definition captures the essential sign of x, assigning discrete values that reflect whether the number is positive, negative, or zero. An equivalent formulation expresses the sign function in terms of the absolute value: \operatorname{sgn}(x) = \frac{x}{|x|} for x \neq 0, with \operatorname{sgn}(0) = 0. This expression highlights its dependence on the magnitude-normalized direction of x, excluding the origin where it is conventionally zero to maintain consistency. The term "signum" is derived from the Latin word for "sign." The modern notation and use of the sign function were introduced by the German mathematician in the late 19th century. In one-dimensional mathematical contexts, the sign function acts as an indicator of direction or orientation, distinguishing the positive and negative halves of the real line while neutralizing zero. This property makes it fundamental for analyzing symmetry, polarity, and stepwise behaviors in real analysis.

Notation and Conventions

The sign function is most commonly denoted using the operator name \operatorname{sgn}(x) or the spelled-out form \operatorname{sign}(x) in mathematical texts, where x is a real number. This notation emphasizes the function's role in extracting the sign of its argument, with \operatorname{sgn} being the more compact and widely adopted variant in analysis and related fields. Alternative notations and expressions arise in specific contexts, such as the relation to the Heaviside step function H(x), where \operatorname{sgn}(x) = 2H(x) - 1 for x \neq 0, providing a piecewise construction that aligns the sign function with indicator-like behaviors in integration and distribution theory. Conventions regarding the value at zero vary across fields and historical periods; in contemporary real analysis, \operatorname{sgn}(0) = 0 is the standard definition to ensure consistency with properties like multiplicativity and to facilitate applications in limits and derivatives. However, some older mathematical treatments leave \operatorname{sgn}(0) undefined, focusing solely on the behavior for nonzero x to highlight the function's discontinuity at the origin. This distinction influences usage in rigorous proofs versus computational implementations. In typesetting, the notation is rendered using mathematical operator commands, such as \operatorname{sgn} in LaTeX, to distinguish it as a function symbol rather than plain text; no dedicated Unicode character exists for "sgn" itself, relying instead on standard Latin letters within mathematical delimiters for clarity in digital and print media.

Fundamental Properties

Algebraic Characteristics

The sign function demonstrates multiplicativity as a key algebraic property. For all real numbers x and y, it satisfies the relation \sgn(xy) = \sgn(x) \sgn(y). This holds because multiplying two positive numbers yields a positive product, two negatives yield a positive product, and one positive and one negative yield a negative product, mirroring the multiplication of their respective signs \pm 1. In contrast, the sign function lacks additivity. Generally, \sgn(x + y) \neq \sgn(x) + \sgn(y). For instance, taking x = 1 and y = -2 gives \sgn(1 + (-2)) = \sgn(-1) = -1, whereas \sgn(1) + \sgn(-2) = 1 + (-1) = 0. Such discrepancies occur because addition can cause sign changes or cancellations that the simple sum of signs fails to capture, highlighting the function's nonlinear nature under addition. The sign function has the property that [\sgn(x)]^2 = 1 for all x \neq 0, and [\sgn(0)]^2 = 0. This arises directly from the range of the function, where non-zero inputs map to \pm 1, and the square of either is 1, while zero maps to itself under squaring. Furthermore, the sign function inherently preserves sign information for expressions. The value of \sgn(f(x)) directly indicates the sign of f(x): positive if f(x) > 0, negative if f(x) < 0, and zero if f(x) = 0.

Relation to Absolute Value

The sign function, denoted \operatorname{sgn}(x), provides a fundamental decomposition of any real number x into its sign and magnitude components. Specifically, for all real x, the identity x = \operatorname{sgn}(x) \cdot |x| holds, where |x| denotes the absolute value of x. This decomposition separates the directional aspect (captured by \operatorname{sgn}(x), which is -1, $0, or $1) from the non-negative magnitude |x|. Conversely, the absolute value can be reconstructed from x and its sign for x \neq 0 via |x| = x / \operatorname{sgn}(x). This relation follows directly from the definition of \operatorname{sgn}(x) = x / |x| for x \neq 0, ensuring the operations are inverses in this context. To verify the equivalence, consider that multiplying \operatorname{sgn}(x) \cdot |x| yields x for x > 0 (since \operatorname{sgn}(x) = [1](/page/1)), -|x| for x < 0 (since \operatorname{sgn}(x) = [-1](/page/−1)), and $0 for x = [0](/page/0). This uniquely recovers x because |x| preserves the magnitude while \operatorname{sgn}(x) restores the original sign, with no other pair of sign and magnitude values producing the same result. This decomposition extends naturally to vectors in \mathbb{R}^n, where the sign function applies component-wise: for \mathbf{x} = (x_1, \dots, x_n), \operatorname{sgn}(\mathbf{x}) = (\operatorname{sgn}(x_1), \dots, \operatorname{sgn}(x_n)) and |\mathbf{x}| = (|x_1|, \dots, |x_n|), satisfying \mathbf{x} = \operatorname{sgn}(\mathbf{x}) \odot |\mathbf{x}| with \odot denoting element-wise multiplication. In this setting, the relation connects to vector norms, such as the \ell_1-norm \|\mathbf{x}\|_1 = \sum_{i=1}^n |x_i|, which aggregates the magnitudes while the sign vector encodes directional information across components. The uniqueness follows analogously, as component-wise recovery ensures no ambiguity in reconstructing \mathbf{x}.

Algebraic Identities

Identities Involving Products

One fundamental identity involving the product of sign functions is the multiplicative property: for all real numbers x and y, \sgn(xy) = \sgn(x) \sgn(y). This holds even when one or both arguments are zero, as \sgn(0) = 0 ensures the product is zero in those cases. To prove this identity, consider cases based on the signs of x and y. If x > 0 and y > 0, then xy > 0, so \sgn(xy) = 1 = (1)(1) = \sgn(x) \sgn(y). If x > 0 and y < 0, then xy < 0, so \sgn(xy) = -1 = (1)(-1) = \sgn(x) \sgn(y). If x > 0 and y = 0, then xy = 0, so \sgn(xy) = 0 = (1)(0) = \sgn(x) \sgn(y). The cases x < 0 and y > 0, x < 0 and y < 0, x < 0 and y = 0, x = 0 and y > 0, x = 0 and y < 0, and x = 0 and y = 0 follow similarly, yielding \sgn(xy) = -1, $1, $0, $0, $0, and $0, respectively, matching \sgn(x) \sgn(y) in each instance. A related identity concerns quotients: for real numbers x and y with y \neq 0, \sgn\left(\frac{x}{y}\right) = \frac{\sgn(x)}{\sgn(y)} = \sgn(x) \sgn(y). The second equality follows because \sgn(y)^2 = 1 for y \neq 0, so $1 / \sgn(y) = \sgn(y). This can be derived from the product identity by noting that x / y = x \cdot (1/y) and \sgn(1/y) = \sgn(y) for y \neq 0, since the reciprocal preserves the sign of a nonzero real number; thus, \sgn(x/y) = \sgn(x) \sgn(1/y) = \sgn(x) \sgn(y). For compositions of functions, under the assumption that the outer function f is differentiable and monotonic (so f' has constant sign), the sign of the composition satisfies \sgn(f(g(x))) = \sgn(f'(g(x))) \sgn(g(x)) for points where g(x) \neq 0 and the expressions are defined. Here, \sgn(f'(g(x))) is constant (+1 if f is strictly increasing, -1 if strictly decreasing), reflecting whether the composition preserves or reverses the sign of g(x). This algebraic relation leverages the product identity applied to the local behavior determined by the derivative's sign./03%3A_Differentiation/3.05%3A_Monotonicity_and_Concavity) These product and quotient identities extend naturally to determining the sign of rational functions. For a rational function r(x) = p(x)/q(x) where p and q are polynomials with q(x) \neq 0, the sign is given by \sgn(r(x)) = \sgn(p(x)) \sgn(q(x)), since \sgn(1/q(x)) = \sgn(q(x)). Each polynomial can be factored into linear terms, and repeated application of the product rule yields the overall sign as the product of the signs of the factors, adjusted for multiplicities (noting that even powers contribute positively via \sgn(z^{2k}) = 1 for z \neq 0, though power-specific details are addressed elsewhere). This approach is essential for sign charts in algebraic analysis, enabling determination of intervals where r(x) is positive or negative without evaluating the full expression.

Identities with Powers and Exponents

The sign function interacts with powers in a manner determined by the parity of the exponent. For a positive even integer exponent $2k where k is a positive integer, \operatorname{sgn}(x^{2k}) = 1 for all real x \neq 0. This holds because x^{2k} = (x^k)^2 > 0 when x \neq 0, and \operatorname{sgn}(y) = 1 for any positive real y. At x = [0](/page/0), \operatorname{sgn}(0^{2k}) = \operatorname{sgn}([0](/page/0)) = 0. In contrast, for a positive odd integer exponent $2k+1, \operatorname{sgn}(x^{2k+1}) = \operatorname{sgn}(x) for all real x \neq 0. This identity follows from expressing the power as a product: x^{2k+1} = x^{2k} \cdot x, so \operatorname{sgn}(x^{2k+1}) = \operatorname{sgn}(x^{2k}) \cdot \operatorname{sgn}(x) = 1 \cdot \operatorname{sgn}(x) = \operatorname{sgn}(x), leveraging the multiplicative property of the sign function. At x = 0, \operatorname{sgn}(0^{2k+1}) = 0 = \operatorname{sgn}(0). A general form for positive integer exponents n is \operatorname{sgn}(x^n) = [\operatorname{sgn}(x)]^n for x \neq 0, derived from \operatorname{sgn}(x^n) = x^n / |x^n| = x^n / |x|^n = (x / |x|)^n = [\operatorname{sgn}(x)]^n. For even n, [\operatorname{sgn}(x)]^n = (\pm 1)^n = 1; for odd n, it equals \operatorname{sgn}(x). This aligns with the even and odd cases above. Regarding roots, the sign of the principal real nth root \sqrt{x} = x^{1/n} depends on whether n is odd or even. For odd positive integer n, \operatorname{sgn}(x^{1/n}) = \operatorname{sgn}(x) for all real x, as the root preserves the sign (negative inputs yield negative outputs, positive yield positive). For even positive integer n, the principal real root is defined only for x \geq 0 and is non-negative, so \operatorname{sgn}(x^{1/n}) = 1 for x > 0 and \operatorname{sgn}(0^{1/n}) = 0; it is undefined in the reals for x < 0. The formal expression \operatorname{sgn}(x)^{1/n} yields 1 for x > 0, but for x < 0 and even n, (-1)^{1/n} involves complex principal branches and is not real. Exponential relations with the sign function are straightforward due to the range of the exponential. Specifically, \operatorname{sgn}(e^x) = 1 for all real x, since e^x > 0 for every real input. Conversely, \operatorname{sgn}(\ln x) is undefined for x \leq 0 because the natural logarithm is only defined for positive ; for x > 0, it equals 1 if x > 1, -1 if $0 < x < 1, and 0 if x = 1. If considering \ln |x| instead, the function is defined for x \neq 0, but the sign follows the same pattern based on whether |x| exceeds, equals, or is less than 1.

Analytic Properties

Discontinuity and Continuity

The sign function, denoted \operatorname{sgn}(x), is continuous at every point a \neq 0 in its domain. For such a, the limit \lim_{x \to a} \operatorname{sgn}(x) = \operatorname{sgn}(a), as the function is constant in any neighborhood excluding zero. This follows from the piecewise constant nature of \operatorname{sgn}(x), which equals 1 for x > 0 and -1 for x < 0. At x = 0, however, \operatorname{sgn}(x) exhibits a jump discontinuity. The right-hand limit is \lim_{x \to 0^+} \operatorname{sgn}(x) = 1, while the left-hand limit is \lim_{x \to 0^-} \operatorname{sgn}(x) = -1, so the two-sided limit \lim_{x \to 0} \operatorname{sgn}(x) does not exist. With \operatorname{sgn}(0) = 0, the function value at zero lies between but does not match either one-sided limit. This discontinuity is classified as a jump rather than removable because the one-sided limits differ from each other; a removable discontinuity requires the one-sided limits to exist and agree, allowing redefinition at the point to restore continuity, which is impossible here. Despite this isolated discontinuity, \operatorname{sgn}(x) remains Riemann integrable over any closed bounded interval [a, b] containing zero, as it is bounded and discontinuous at only a finite number of points. The integral can be computed piecewise, yielding $0 if a < 0 < b with |a| = b, for example, due to the symmetric cancellation from the positive and negative parts.

Smooth Approximations

The sign function's discontinuity at the origin limits its use in contexts requiring differentiability, such as , numerical simulations, and . Smooth approximations mitigate this by constructing infinitely differentiable functions that replicate the sign function's behavior for large |x| while providing a gradual transition near x = 0, controlled by a sharpness parameter k > 0. A prominent smooth approximation employs the : \sgn(x) \approx \tanh(kx) as k → ∞. The hyperbolic tangent is strictly increasing, odd, and saturates at ±1, making it suitable for emulating the sign function's range and symmetry. This form arises naturally in applications like neural networks, where tanh serves as a differentiable surrogate for binary decisions. As k increases, tanh(kx) converges pointwise to sgn(x) for all real x, since lim_{k→∞} tanh(kx) = sgn(x). Another common approximation utilizes the , defined as \erf(z) = \frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} \, dt, which is also and approaches ±1 asymptotically. A normalized form is \sgn(x) \approx \erf\left( \frac{k \sqrt{\pi}}{2} x \right) for large k. The factor \frac{\sqrt{\pi}}{2} ensures the at x = 0 equals k, aligning the transition sharpness with that of the tanh approximation and facilitating consistent error comparisons across methods. This Gaussian-based smoothing is particularly useful in probabilistic models and processes. Logistic functions provide yet another family of approximations, based on the σ(y) = 1 / (1 + e^{-y}). The centered version \sgn(x) \approx 2 \sigma(kx) - 1 converges to sgn(x) as k → ∞ and is mathematically equivalent to tanh(kx/2), inheriting similar saturation properties. Logistic approximations are favored in statistical modeling and for their interpretability in terms of cumulative probabilities. These approximations exhibit to sgn(x) everywhere, but the rate slows near x = 0 due to the enforced smoothness. Uniform convergence holds on compact sets excluding the , such as [-M, -δ] ∪ [δ, M] for δ > 0 and M > 0: the supremum error sup_{x ∈ K} |approx(kx) - sgn(x)| → 0 as k → ∞, since away from zero, the approximants approach the constant ±1 uniformly by of the on the . Near zero, the error is bounded by O(1/k), reflecting the width of the transition region, which vanishes inversely with k but prevents global uniform convergence.

Differentiation

The signum function \sgn(x) is differentiable everywhere except at x = 0, where its does not exist in the classical sense. For x \neq 0, \sgn(x) is locally constant, so its classical is \sgn'(x) = 0. At x = 0, the right-hand is \frac{\sgn(h) - \sgn(0)}{h} = \frac{1}{h} for h > 0, which tends to +\infty as h \to 0^+, and the left-hand is \frac{-1}{h} for h < 0, which also tends to +\infty as h \to 0^-. Since both one-sided limits are infinite rather than finite, the at zero is undefined classically. In the sense of distributions (or weak derivatives), the signum function is differentiable everywhere, with its distributional derivative given by \sgn'(x) = 2 \delta(x), where \delta(x) is the Dirac delta distribution. This follows from the relation \sgn(x) = 2 H(x) - 1, where H(x) is the Heaviside step function whose distributional derivative is the Dirac delta \partial_x H(x) = \delta(x). For compositions, if f: \mathbb{R} \to \mathbb{R} is a smooth function, the distributional derivative of \sgn(f(x)) is \frac{d}{dx} [\sgn(f(x))] = 2 f'(x) \delta(f(x)). This is a consequence of the chain rule for distributional derivatives applied to the composition of the signum function with a smooth map.

Integration

The antiderivative of the sign function \operatorname{sgn}(x) is |x| + C, where C is the constant of integration. This follows from the fact that the derivative of |x| yields \operatorname{sgn}(x) for x \neq 0, and the function is continuous everywhere, allowing the fundamental theorem of calculus to apply away from the origin. For definite integrals over intervals that include zero, the sign function's piecewise constant nature simplifies computation by splitting the integral at the origin. Specifically, for a > 0 and b > 0, \int_{-a}^{b} \operatorname{sgn}(x) \, dx = \int_{-a}^{0} (-1) \, dx + \int_{0}^{b} 1 \, dx = -a + b = b - a. This result highlights the function's contribution as the net "signed length" across the positive and negative domains. Improper integrals involving the sign function over infinite domains often require the when the integrand does not converge absolutely. For instance, the principal value \int_{-\infty}^{\infty} \frac{\operatorname{sgn}(x)}{1 + x^2} \, dx = 0, since the integrand is —the product of the odd \operatorname{sgn}(x) and the even $1/(1 + x^2)—and the principal value of an function over symmetric limits vanishes when the limits exist. The sign function is Lebesgue integrable over any finite interval [c, d] because it is bounded (|\operatorname{sgn}(x)| \leq 1) and the interval has finite Lebesgue measure, ensuring the integral of its absolute value remains finite. This contrasts with its non-differentiability at zero but aligns with the broader theory of integration for measurable functions on sets of finite measure.

Fourier Transform

The of the sign function \sgn(x) in one dimension is given in the principal value sense by \mathcal{F}\{\sgn(x)\}(\omega) = \pv \frac{2}{i \omega}, where the principal value addresses the at \omega = 0. This expression arises from the convention where the is defined as \mathcal{F}\{f\}(\omega) = \int_{-\infty}^{\infty} f(x) e^{-i \omega x} \, dx, adjusted for the non-integrability of \sgn(x). The result can be derived by considering the of \sgn(x), which is $2\delta(x), and applying the differentiation property of the , yielding the form up to a determined by and test functions. Since \sgn(x) is not absolutely integrable, the transform does not exist in the classical Lebesgue sense but converges conditionally as a tempered distribution. To establish convergence, limiting procedures such as (averaging partial integrals) or Abel summation (using exponential damping factors like e^{-\epsilon |x|} and taking \epsilon \to 0) are employed, ensuring the integral is well-defined in the distributional . These methods regularize the oscillatory behavior at and the at zero, making the transform usable in . In signal processing, the sign function relates directly to the Hilbert transform, where \sgn(\omega) serves as the frequency-domain multiplier (up to a factor of -i) that implements a 90-degree phase shift for positive frequencies and -90 degrees for negative ones. Specifically, the Hilbert transform \mathcal{H}\{f\}(x) = \pv \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{f(t)}{x - t} \, dt equals the convolution f(x) * \frac{1}{\pi x}, and its Fourier transform is -i \sgn(\omega) \hat{f}(\omega). This property enables applications such as extracting the analytic signal from real-valued inputs by suppressing negative frequencies, crucial for amplitude modulation and envelope detection. The principal value ensures the transform handles discontinuities appropriately in these contexts.

Generalizations and Extensions

Complex Signum Function

The complex signum function extends the signum function to complex numbers z \in \mathbb{C}. It is defined as \sgn(z) = 0 if z = 0, and \sgn(z) = \frac{z}{|z|} if z \neq 0, where |z| denotes the of z. Equivalently, \sgn(z) = e^{i \arg(z)} for z \neq 0, expressing the unit complex number with the same as z. The principal branch of the complex signum function employs the principal argument \Arg(z) \in (-\pi, \pi], introducing a branch cut along the negative real axis (including the origin) to ensure single-valuedness. This choice aligns with the standard convention for the argument function, where the cut prevents the multi-valued nature of the argument from affecting continuity in the slit plane. The principal branch is continuous in \mathbb{C} minus the non-positive real axis, but discontinuous across the branch cut and at the origin, and is not holomorphic anywhere. The complex signum relates directly to the via the z = \sgn(z) |z| for all z \in \mathbb{C}, decomposing any nonzero into its directional (phase) and components.

Matrix Sign

The matrix sign , denoted \operatorname{sgn}(A), extends the scalar signum to square matrices A \in \mathbb{C}^{n \times n} via the . For a A = V D V^{-1}, where D = \operatorname{diag}(\lambda_1, \dots, \lambda_n) contains the eigenvalues of A, the matrix sign is defined by \operatorname{sgn}(A) = V \operatorname{sgn}(D) V^{-1}, with \operatorname{sgn}(D) = \operatorname{diag}(\operatorname{sgn}(\lambda_1), \dots, \operatorname{sgn}(\lambda_n)) and \operatorname{sgn}(\lambda) = \lambda / |\lambda| for \lambda \neq 0. This assumes no eigenvalues lie on the imaginary , ensuring the scalar signum is holomorphic in a neighborhood of the of A. The extends the to non-diagonalizable matrices, yielding a unique \operatorname{sgn}(A) that satisfies the same spectral mapping property: the eigenvalues of \operatorname{sgn}(A) are \operatorname{sgn}(\lambda_i). Key properties of the matrix sign function mirror those of the scalar case, adapted to the matrix setting. For an invertible matrix A (with no zero eigenvalues), \operatorname{sgn}(A)^2 = I_n, the n \times n , since the eigenvalues of \operatorname{sgn}(A) have modulus 1 and square to 1. Additionally, if A and B are invertible square matrices that commute (i.e., AB = BA), then \operatorname{sgn}(AB) = \operatorname{sgn}(A) \operatorname{sgn}(B), reflecting the multiplicative nature of the scalar signum under suitable conditions. These properties hold more generally for matrices without pure imaginary or zero eigenvalues, facilitating applications in numerical linear algebra and control theory. The matrix sign function provides the unitary (or partial ) factor in the polar decomposition of A. Specifically, any A admits a unique A = U H, where U = \operatorname{sgn}(A) is unitary if A is invertible (or a partial otherwise) and H = |A| = (A^* A)^{1/2} is the unique of A^* A, with A^* denoting the . This connection links the sign function to techniques, where \operatorname{sgn}(A) separates the "direction" from the "magnitude" of A, analogous to the scalar decomposition z = \operatorname{sgn}(z) |z|. For real matrices, the decomposition yields an orthogonal factor when applicable. A primary method for computing \operatorname{sgn}(A) is the Newton iteration, defined by X_0 = A and X_{k+1} = \frac{1}{2} (X_k + X_k^{-1}) for k \geq 0. This iteration converges quadratically to \operatorname{sgn}(A) provided A has no eigenvalues on the imaginary axis, with the error satisfying \|X_{k+1} - \operatorname{sgn}(A)\| \leq C \|X_k - \operatorname{sgn}(A)\|^2 for some constant C > 0 near convergence. The method is globally convergent under the spectral condition and requires solving linear systems at each step, making it efficient for moderate-sized matrices; scaling techniques can enhance stability for ill-conditioned A. This approach, rooted in early work on matrix iterations, remains a cornerstone for practical implementations in software libraries.

Signum in the Sense of Distributions

The signum function defines a regular distribution on the space of test functions C_c^\infty(\mathbb{R}) via its local integrability, with action given by \langle \sgn, \phi \rangle = \int_{-\infty}^{\infty} \sgn(x) \phi(x) \, dx for \phi \in C_c^\infty(\mathbb{R}). The distributional derivative of \sgn is computed using the definition \langle \sgn', \phi \rangle = -\langle \sgn, \phi' \rangle, which yields -\int_{-\infty}^{\infty} \sgn(x) \phi'(x) \, dx = -\left( \int_0^{\infty} \phi'(x) \, dx - \int_{-\infty}^0 \phi'(x) \, dx \right) = - \left( (0 - \phi(0)) - (\phi(0) - 0) \right) = 2\phi(0). This equals \langle 2\delta, \phi \rangle, so \sgn' = 2\delta in the distributional sense, where \delta is the Dirac delta distribution. The computation relies on integration by parts across the discontinuity at zero, confirming the jump of magnitude 2 contributes the delta factor. Higher-order distributional derivatives follow inductively from the properties of distributional differentiation. Differentiating \sgn' = 2\delta gives \sgn'' = 2\delta', and continuing yields \sgn^{(n)} = 2 \delta^{(n-1)} for all integers n \geq 1. For odd n, these derivatives are even distributions, aligning with the odd parity of \sgn itself. The of the signum , defined for tempered , is obtained using the differentiation property: the transform of \sgn' = 2\delta implies i\omega \hat{\sgn}(\omega) = 2 \hat{1}, adjusted for conventions where \hat{\delta} = 1. Accounting for the of \sgn (which forces the transform to be purely imaginary and ), the result is the principal value \hat{\sgn}(\omega) = -\frac{i}{\pi \omega} \ \mathrm{p.v.}, where the principal value handles the singularity at \omega = 0.