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Expression Search

The following examples illustrate searches for mathematical expressions using standard LaTeX notation. Depending on the query and search settings, matches may correspond to complete expressions or to structurally meaningful parts of larger expressions.

• a^2+b^2=c^2 (Options: Normalize, Confidence=50)
• E=mc^2 (Options: Normalize, Confidence=50)
• \sqrt{x^2+y^2} (Options: Normalize, Confidence=50)
• \frac{n(n+1)}{2} (Options: Normalize, Confidence=50)
• \frac{n(n-1)}{2} (Options: Normalize, Confidence=50)
• \Omega_M=\frac{1}{2}\Omega_{M[AB]}\Sigma^{AB} (Options: Normalize, Confidence=50)
• \frac{\mu_k-\bar{\mu}_k}{\zeta-\mu_k} (Options: Normalize, Confidence=50)
• \frac{B^2}{A^2}-\frac{l^2}{r^2} (Options: Normalize, Confidence=50)
• \int_0^\infty d^D r_\alpha V(r_\alpha) (Options: Normalize, Confidence=50)
• AdS_3 \times S^3 \times S^3 \times S^1 (Options: Normalize, Confidence=50)

Segmentation-Based Search

These examples demonstrate searches that use the Segment option to automatically decompose the query into structurally meaningful sub-expressions. When enabled, lower coverage values increase recall by accepting partial matches, whereas higher values require a closer correspondence to the original query.

• \Delta=(\begin{array}{c} x-i \lambda 1_2 \\ x+i \lambda 1_2 \end{array}) (Options: Normalize, Segment, Min.Tokens=2, Segm.Coverage=0, Confidence=50)
• \{ \begin{array}{c} v_t = -u_x \\ u_t = -v^{n-2} v_{x} \end{array} (Options: Normalize, Segment, Min.Tokens=2, Segm.Coverage=40, Confidence=50)
• \gamma_A = \frac{ \sqrt{2} C(\kappa) \alpha} {24\pi (2\kappa^2-1)^3} (Options: Normalize, Segment, Min.Tokens=2, Segm.Coverage=30, Confidence=50)
• ( \begin{array}{cc} a1 & b\eta \\ c\eta & d1 \end{array} ), ad-bc=1 (Options: Normalize, Segment, Min.Tokens=2, Segm.Coverage=20, Confidence=50)
• H_{2m+1} = -\frac{1}{(2m+1) \kappa^m} sTr L^{\frac{2m+1}{2}} (Options: Normalize, Segment, Min.Tokens=2, Segm.Coverage=20, Confidence=50)
• {( \frac{\dot {a}}{a})} ^2 = \frac{8\pi G}{3} \rho -\frac{k}{a^2} (Options: Normalize, Segment, Min.Tokens=2, Segm.Coverage=30, Confidence=50)
• Z[s(\sigma)] = {[\underset{n\rightarrow\infty}{\operatorname{lim}} {([s(\sigma)]^{-D \slash 2} e^{ -\frac{m^{2}}{2} \int d^p \sigma s(\sigma)} V )} ^n]}^N (Options: Normalize, Segment, Min.Tokens=2, Segm.Coverage=30, Confidence=0)
• (\frac {\partial} {\partial \chi} - r + \sigma) (\begin{array}{c} a \\ b \end{array}) = (\begin{array}{c} - (\hat{\phi} - \zeta) h \\ 2 \bar {\partial}_{\hat{A} + \hat{a}} h \end{array}) (Options: Normalize, Segment, Min.Tokens=2, Segm.Coverage=20, Confidence=0)

Boolean Queries: OR

The OR operator returns pages matching any of the specified query expressions. When Segment and Segm.Coverage are used with OR queries, coverage is computed across the combined decomposed segments but requires those matched segments to occur within the same expression. This can be useful, but the interpretation may be ambiguous because matching segments may come from different OR branches.

• n(n+1) OR N(N+1) (Options: Normalize, Confidence=50)
• \sin\theta OR \cos\theta (Options: Normalize, Confidence=50)
• e^{i\vartheta}\sinh\tau OR e^{-i\vartheta}\sinh\tau (Options: Normalize, Confidence=50)
• \frac{n(n-1)}{2} OR \frac{N(N-1)}{2} (Options: Normalize, Segment, Min.Tokens=2, Segm.Coverage=40, Confidence=50)
• \begin{array}{cc} 0 & I \\ - I & 0 \end{array} OR \begin{array}{cc} 0 & I d \\ - I d & 0 \end{array}(Options: Normalize, Segment, Min.Tokens=1, Segm.Coverage=40, Confidence=50)

Boolean Queries: AND

Match pages containing all specified query components. When Segment and Segm.Coverage are used with AND queries, coverage is computed across the combined decomposed segments but requires those matched segments to occur within the same expression. This can be useful, but the interpretation may be ambiguous because matching segments may come from different AND branches.

• a^2 + b^2 AND c^2 (Options: Normalize, Confidence=50)
• \sin\theta AND \cos\theta (Options: Normalize, Confidence=50)
• e^{i\vartheta}\sinh\tau AND \cosh\tau (Options: Normalize, Confidence=50)
• e^{i\vartheta}\sinh\tau AND e^{-i\vartheta}\sinh\tau (Options: Normalize, Confidence=20)
• \frac{n(n-1)}{2} AND \frac{n(n+1)}{2} (Options: Normalize, Confidence=50)
• \begin{array}{cc} A & B \\ C & D \end{array} AND SO(2,2) (Options: Normalize, Segment, Min.Tokens=1, Segm.Coverage=60, Confidence=50)

Boolean Queries: NOT

The NOT operator returns pages matching the expression on the left, while reducing or excluding pages that also match the expression on the right. This exclusion is confidence-based: a weak match on the right may only lower the score and may not remove the page. The right-hand expression must match strongly enough, relative to the selected Confidence threshold, to act as an effective veto. Segm.Coverage is currently disabled for NOT queries because coverage mode does not yet support exclusion branches.

• \frac{n(n+1)}{2} NOT \frac{n(n-1)}{2} (Options: Normalize, Confidence=50)
• \sin\theta NOT \cos\theta (Options: Normalize, Confidence=50)
• \Omega_M NOT \Sigma^{AB} (Options: Normalize, Confidence=50)

Grouping and Nested Queries

Parentheses allow multiple Boolean conditions to be combined into a single query. Grouped expressions are evaluated at the page level, making it possible to search for pages satisfying complex logical combinations of mathematical expressions.

• e^{i\vartheta}\sinh\tau $AND $APAR \cosh\tau $OR \sinh\tau $CPAR (Options: Normalize, Confidence=50)
• $APAR \frac{1}{2} OR M[AB] $CPAR AND \Omega_M (Options: Normalize, Confidence=50)

Sequence Constraints (SEQ)

Search for expressions whose components appear in a specified symbolic order. SEQ is a more restrictive form of the AND operator. All query expressions or segments must be matched within the same mathematical expression and must appear in the order specified by the query. The ordering is determined by the canonical reading order of the mathematical expression rather than by their visual layout on the page. Sequence queries must be enclosed within $ACOR and $CCOR delimiters, which can be inserted using the corresponding bracket buttons in the query editor.

• $ACOR $APAR \sin OR \cos $CPAR $MCOL \alpha $CCOR (Options: Normalize, Confidence=50)
• $ACOR $APAR e^{i\vartheta}\sinh\tau OR e^{-i\vartheta}\sinh\tau $CPAR $MCOL \cosh\tau $CCOR (Options: Normalize, Confidence=20)

Query-by-Image

Search using a cropped image of a mathematical expression instead of typing LaTeX.

• Upload an image containing a mathematical expression.
• Select a formula directly from a document page using Recognize from crop.
• Edit the recognized expression before searching.