add more help text

cmd/convergents.go

@@ -49,7 +49,12 @@ func convergents(cmd *cobra.Command, args []string) {
 var convergentsCmd = &cobra.Command{
 	Use:   "convergents N N [N ...]",
 	Short: "Compute convergents of a periodic continued fraction",
-	Long:  `Compute convergents of a periodic continued fraction.`,
+	Long: `Compute convergents of a periodic continued fraction.
+
+The first number given is the integer part. The following numbers give the repetend of the continued fraction.
+
+This will output convergents infinitely. Try piping to head to only output a certain number of convergents, like this:
+mathtools convergents 3 2 6 | head -n 5`,
 	Args: cobra.MinimumNArgs(2),
 	Run:  convergents,
 }

cmd/discreteLog.go

@@ -119,9 +119,13 @@ func discreteLog(cmd *cobra.Command, args []string) {
 
 // discreteLogCmd represents the discreteLog command
 var discreteLogCmd = &cobra.Command{
-	Use:   "discrete-log",
+	Use:   "discrete-log -b N -m N -e N",
 	Short: "Compute the discrete logarithm",
-	Long:  `Compute the discrete logarithm.`,
+	Long: `Compute the discrete logarithm.
+
+Given a base b, modulus m, and element e, compute a value k such that b^k = e (mod m).
+
+Note that no efficient method of finding the discrete logarithm is currently known. For slightly improved performance, the order of the group (i.e. the totient of m) can be provided.`,
 	Run: discreteLog,
 }
 

cmd/divisor.go

@@ -59,7 +59,10 @@ func divisorSum(cmd *cobra.Command, args []string) {
 var divisorCmd = &cobra.Command{
 	Use:   "divisor N [N ...]",
 	Short: "Compute the divisor summatory function",
-	Long:  `Compute the divisor summatory function.`,
+	Long: `Compute the divisor summatory function.
+
+For each argument n, compute D(n) = d(1) + d(2) + d(3) + ... + d(n), where d(n) is the number of divisors of n.`,
+	Args: cobra.MinimumNArgs(1),
 	Run:  divisorSum,
 }
 

cmd/divisors.go

@@ -103,9 +103,18 @@ func divisors(cmd *cobra.Command, args []string) {
 
 // divisorsCmd represents the divisors command
 var divisorsCmd = &cobra.Command{
-	Use:   "divisors",
+	Use:   "divisors -n N",
 	Short: "Compute the divisor function for all numbers less than n",
-	Long:  `Compute the divisor function for all numbers less than n.`,
+	Long: `Compute the divisor function for all numbers less than n.
+
+sigma_x(n) computes the sum of the xth powers of the divisors of n.
+For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12.
+sigma_0(12) = 6
+sigma_1(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28
+sigma_2(12) = 1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 12^2 = 210
+
+This program computes sigma for all values less than the input n.
+Provide the --exponent flag to set the desired exponent for each sum.`,
 	Run: divisors,
 }
 

cmd/gcd.go

@@ -69,7 +69,10 @@ func gcd(cmd *cobra.Command, args []string) {
 var gcdCmd = &cobra.Command{
 	Use:   "gcd N N [N ...]",
 	Short: "Compute the greatest common denominator of a set of numbers",
-	Long:  `Compute the greatest common denominator of a set of numbers.`,
+	Long: `Compute the greatest common denominator of a set of numbers.
+
+Provide the --extended flag to also compute the Bézout coefficients, i.e. coefficients a,b,c,... such that
+ax + by + cz + ... = gcd(x,y,z,...)`,
 	Args: cobra.MinimumNArgs(2),
 	Run:  gcd,
 }
@@ -87,5 +90,5 @@ func init() {
 	// is called directly, e.g.:
 	// gcdCmd.Flags().BoolP("toggle", "t", false, "Help message for toggle")
 
-	gcdCmd.Flags().BoolVarP(&extended, "extended", "e", false, "also compute the Bezout coefficients")
+	gcdCmd.Flags().BoolVarP(&extended, "extended", "e", false, "also compute the Bézout coefficients")
 }

cmd/jacobi.go

@@ -51,7 +51,15 @@ func jacobi(cmd *cobra.Command, args []string) {
 var jacobiCmd = &cobra.Command{
 	Use:   "jacobi -a A -n N",
 	Short: "Compute the Jacobi symbol",
-	Long:  `Compute the Jacobi symbol (a | n).`,
+	Long: `Compute the Jacobi symbol (a | n).
+
+The Jacobi symbol is useful for studying quadratic residues. a is a quadratic residue modulo n if there exists a value x such that
+x^2 = a (mod n)
+
+If a is a quadratic residue modulo n and gcd(a, n) = 1, then (a | n) = 1.
+If (a | n) = -1, then a is a quadratic nonresidue modulo n.
+
+However, (a | n) = 1 does not guarantee that a is a quadratic residue modulo n.`,
 	Run: jacobi,
 }
 

cmd/mobius.go

@@ -73,9 +73,15 @@ func mobius(cmd *cobra.Command, args []string) {
 
 // mobiusCmd represents the mobius command
 var mobiusCmd = &cobra.Command{
-	Use:   "mobius",
+	Use:   "mobius -n N",
 	Short: "Compute the Möbius function for all numbers less than n",
-	Long:  `Compute the Möbius function for all numbers less than n.`,
+	Long: `Compute the Möbius function for all numbers less than n.
+
+The Möbius function mu(n) is a multiplicative function defined as follows for prime p:
+mu(p) = -1
+mu(p^k) = 0 for k > 1
+
+This means for squarefree n, mu(n) = -1 if n has an odd number of prime factors and mu(n) = 1 if n has an even number of prime factors. mu(n) = 0 if n is not squarefree.`,
 	Run: mobius,
 }
 

cmd/modInverse.go

@@ -48,9 +48,11 @@ func modInverse(cmd *cobra.Command, args []string) {
 
 // modInverseCmd represents the modInverse command
 var modInverseCmd = &cobra.Command{
-	Use:   "mod-inverse",
+	Use:   "mod-inverse -g N -m N",
 	Short: "Compute a modular inverse",
-	Long:  `Compute a modular inverse.`,
+	Long: `Compute a modular inverse.
+
+Given a base g and modulus m, compute a value x such that gx = 1 (mod m). A solution only exists if g and m are coprime.`,
 	Run: modInverse,
 }
 

cmd/partitions.go

@@ -89,9 +89,19 @@ func partitions(cmd *cobra.Command, args []string) {
 
 // partitionsCmd represents the partitions command
 var partitionsCmd = &cobra.Command{
-	Use:   "partitions",
+	Use:   "partitions -n N -k N",
 	Short: "Compute the number of partitions of an integer",
-	Long:  `Compute the number of partitions of an integer.`,
+	Long: `Compute the number of partitions of an integer.
+
+For example, 4 can be partitioned into
+4
+3+1
+2+2
+2+1+1
+1+1+1+1
+so p(n) = 5.
+
+To compute the number of partitions into at most k parts, provide the -k flag.`,
 	Run: partitions,
 }
 

cmd/pell.go

@@ -60,9 +60,12 @@ func pell(cmd *cobra.Command, args []string) {
 
 // pellCmd represents the pell command
 var pellCmd = &cobra.Command{
-	Use:   "pell",
+	Use:   "pell -d N",
 	Short: "Find solutions to a Pell equation",
-	Long:  `Find solutions to a Pell equation x^2 - dy^2 = 1.`,
+	Long: `Find integer solutions to a Pell equation x^2 - dy^2 = 1.
+
+This will output solutions infinitely. Try piping to head to only output a certain number of solutions, like this:
+mathtools pell -d 12 | head -n 5`,
 	Run: pell,
 }
 

cmd/shoelace.go

@@ -99,9 +99,17 @@ func shoelace(cmd *cobra.Command, args []string) {
 
 // shoelaceCmd represents the shoelace command
 var shoelaceCmd = &cobra.Command{
-	Use:   "shoelace",
+	Use:   "shoelace -f FILE",
 	Short: "Compute the area of a simple polygon from the vertex coordinates",
-	Long:  `Compute the area of a simple polygon from the vertex coordinates.`,
+	Long: `Compute the area of a simple polygon from the vertex coordinates.
+
+Put each point on its own line, with each coordinate separated by whitespace. For example, a file with
+1 6
+3 1
+7 2
+4 4
+8 5
+will output an area of 16.5.`,
 	Run: shoelace,
 }
 

cmd/sqrtRepetend.go

@@ -52,8 +52,15 @@ func sqrtRepetend(cmd *cobra.Command, args []string) {
 // sqrtRepetendCmd represents the sqrtRepetend command
 var sqrtRepetendCmd = &cobra.Command{
 	Use:   "sqrt-repetend N [N ...]",
-	Short: "Compute the repetend of the square root of the input",
-	Long:  `Compute the repetend of the square root of the input.`,
+	Short: "Compute the repetend of the continued fraction of square roots",
+	Long: `Compute the repetend of the continued fraction of the square root of the input.
+
+For each argument n, this will output the repetend of the continued fraction of sqrt(n). For example,
+mathtools sqrt-repetend 12 15 19
+will output
+2 6
+1 6
+2 1 3 1 2 8`,
 	Args: cobra.MinimumNArgs(1),
 	Run:  sqrtRepetend,
 }

cmd/stirling.go

@@ -58,9 +58,23 @@ func stirling(cmd *cobra.Command, args []string) {
 
 // stirlingCmd represents the stirling command
 var stirlingCmd = &cobra.Command{
-	Use:   "stirling",
+	Use:   "stirling [-1|-2] -n N -k N",
 	Short: "Compute the Stirling numbers",
-	Long:  `Compute the Stirling numbers.`,
+	Long: `Compute the Stirling numbers.
+
+The Stirling numbers of the first kind give the coefficients in the expansion of the falling factorial. For example,
+x(x-1)(x-2) = x^3 - 3x^2 + 2x
+Consequently, s(3,1) = 2, s(3,2) = -3, and s(3,3) = 3.
+
+The Stirling numbers of the second kind count the number of ways to partition a set of n objects into k non-empty subsets. For example, there are 7 ways to partition a set of 4 objects into 2 non-empty subsets:
+{1} {2,3,4}
+{2} {1,3,4}
+{3} {1,2,4}
+{4} {1,2,3}
+{1,2} {3,4}
+{1,3} {2,4}
+{1,4} {2,3}
+Therefore S(4,2) = 7.`,
 	Run: stirling,
 }
 

cmd/totient.go

@@ -63,9 +63,11 @@ func totient(cmd *cobra.Command, args []string) {
 
 // totientCmd represents the totient command
 var totientCmd = &cobra.Command{
-	Use:   "totient",
+	Use:   "totient -n N",
 	Short: "Compute the totient function for all numbers less than n",
-	Long:  `Compute the totient function for all numbers less than n.`,
+	Long: `Compute the totient function for all numbers less than n.
+
+The totient function phi(n) counts the numbers up to n that are coprime to n.`,
 	Run: totient,
 }