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yyy/english/ophelp2.c /home/beeson/Otter-Lambda GNU C 3.3.5 (Debian 1:3.3.5-13) �
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yyy/english yyy ophelp2.c operator.h �9� Calculate the function near the limit point, at values you will specify. The limit of a sum is the sum of the limits (if defined). The limit of a difference is the difference of the limits (if defined). Example: $lim(t->3,\pi ) = \pi $ Example: lim(t->3,t) = 3 Pull out a constant through the limit sign. Pull out a minus sign through a limit. The limit of a product is the product of the limits (if defined). The limit of a (constant) power is the power of the limit. Example: lim(x->3,2^x) = 2^lim(x->3,x) The limit of a power is the power of the limits (if defined). Watch out for the case when the limit is zero. It still works if $u\ge 0$. The limit of an odd root is the root of the limit. Watch out for the case when the limit is zero. It still works if $u\ge 0$. Calculate the limit of a polynomial in the limit variable in one step. Example: lim(x->0,|x^3|) = |lim(x->0,x^3| Pull constants out of the numerator and denominator through the limit sign. Applies only if the numerator is constant. Does not work if lim u and lim v are both zero or infinity. Factor powers of (x-a) out of both numerator and denominator, if possible. Calculate the limit of a quotient of polynomials in one step. Use this law to prepare to push the limit through the power. Example: This will multiply num and denom of $(x-1)/(\sqrt x-1)$ by $\sqrt x+1$. Example: in the limit of (x-1)^2 sin x/ tan x as x->0, pull out lim (x-1)^2. $ab + ac = a(b+c)$, where $a$ does not depend on the limit variable. You will be asked what to multiply num and denom by. You will get a limit of a compound fraction, not a quotient of limits. You will get a quotient of limits, not a limit of a compound fraction. Example: use this on $(sin x cos h + cos x sin h - sin x)/h$ Example: $\sqrt x/2 = \sqrt (x/4)$ Example: $\sqrt x/(-2) = -\sqrt (x/4)$ Example: $^3\sqrt a/2 = ^3\sqrt (a/8)$ Example: $^4\sqrt x/(-2) = -^4\sqrt (x/16) (b<0, n even)$ Example: $2/\sqrt x = \sqrt (4/x)$ Example: $(x-1)/\sqrt x = -\sqrt ((x-1)^2/x)$ when $x\le 1$ Example: $2/+^3\sqrt x = ^3\sqrt (8/x)$ Example: $(x-1)/^3\sqrt x = -^3\sqrt (x-1)^n/x)$ when $x\le 1$ Replace an indeterminate limit of a quotient by the limit of the derivatives. Uses all the rules about derivatives to get the answer in one step. Example: lim x ln x = lim (ln x)/(1/x). Then use L'H�pital's rule. Example: $lim x (ln x)^2 = lim (ln x)^2/(1/x)$. Then use L'H�pital's rule. Example: lim x^(-3) e^x = lim e^x/x^3. Example: lim x^3 e^x = lim x^3/e^(-x). Then use L'H�pital's rule. Examples: $lim f(x) tan x = lim f(x)/cot x$; $lim f(x) sin x = lim f(x)/csc x$. You will be asked which factor to move to the denominator. Put fractions over a common denominator and simplify. For small t, sin t is approximately t. For small t, tan t is approximately t. cos t goes to 1 quite rapidly, faster than t goes to zero. cos t goes to 1 like t^2, as t goes to 0. The coefficient is $\onehalf $. For example (1+ .001)^1000 is pretty close to e. For small t, ln(1+t) is approximately t. For small t, e^t-1 is approximately t. Any power of t, even a fractional power, will kill the singularity in ln. cos (1/t) oscillates between -1 and 1 infinitely many times as t0. sin (1/t) oscillates between -1 and 1 infinitely many times as t0. tan (1/t) has large oscillations and is not even everywhere defined near t=0. cos t oscillates between -1 and 1 infinitely many times as t$\infty $. sin t oscillates between -1 and 1 infinitely many times as t$\infty $. tan t has large oscillations and is not even everywhere defined as t$\infty $. For small t, sinh t is approximately t. For small t, tanh t is approximately t. cosh t goes to 1 quite rapidly, faster than t goes to zero. cosh t goes to 1 like t^2, as t goes to zero. The coefficient is $\onehalf $. Push limit through ln. Example: lim sin x^2 = sin lim x^2 lim(ta,f(g(t)))=lim(ug(a),f(u)) Evaluate a limit in one step, if within MathXpert's capabilities. Example: lim x^x as x0 = lim e^(x ln x) You will be asked for the factor to move to the denominator. Example, limit of $\sqrt x$ as x0 is undefined since $\sqrt x$ is not defined for x < 0. Example: $lim x^x = e^(lim ln x^x)$ Example: lim x sin(1/x) as x0 = 0 since $|sin(1/x)| \le 1$ and x0. Rationalize the numerator, except that no fraction is originally present. Drop terms in numerator and denominator which are dominated by other terms. Example: lim (x + x^2 sin x) = lim x as x0 since (x^2 sin x)/x 0 Replaces u+v by u if v/u0. Use with care! See help for explanations. Example: $sin(undefined) = undefined$ Example: $lim e^(1/x) = e^(lim 1/x)$ Push limit through ln For large t, 1/t^n is small. For large t, t^n is large For large t, e^t is large For large negative t, e^t is small. For large t, ln t is large. For large t, $\sqrt t$ is large. For large t, $^n\sqrt t$ is large. The arctan of a large positive (or negative) number is almost $\pi /2$ (or $-\pi /2$). The arccot of a large positive number is close to zero. The arccot of a large negative number is close to $\pi $. tanh of a large positive (or negative) number is almost 1 (or -1). Push limit through sin Push limit through cos $lim(t�,f(t))=lim(t0+,f(1/t))$ Example: $lim 1/t^4 \infty $ as t320 Example: the two-sided limit, lim 1/t^3 as t0, is undefined. Example: the right-hand limit, lim 1/t^3 as t0+, is $\infty $. Example: the left-hand limit, lim 1/t^3 as t0-, is $-\infty $. Example: lim 1/t as t0 is undefined. This one-sided limit is $-\infty $, but the two-sided limit is undefined. The given one-sided limits are $\pm \infty $, but the two-sided limits are undefined. Example: $lim(t->0, ln(1+t) e^t)$ becomes $lim(t->0, ln(1+t)/t) lim(t->0,te^t)$. Example: $lim(t->0,t ln(1+t))$ becomes $lim(t->0, t^2) lim(t->0,ln(1+t)/t)$. Example: $\infty /2 = \infty $ Example: $1/\infty = 0$ Example: $2\times \infty = \infty $ This rule is shorthand for $lim uv = \infty $ if $lim u = \infty $ and $lim v = \infty $. Example: $\infty + 2 = \infty $ This rule is shorthand for $lim u+v = \infty $ if $lim u = \infty $ and $lim v = \infty $. Example: $e^\infty = \infty $ Example: $(\onehalf )^\infty = 0$ Example: $e^(-\infty ) = 0$ Example: $(\onehalf )^(-\infty ) = \infty $ Example: $\infty ^3 = \infty $ You cannot cancel $\infty -\infty $. This expression is undefined. 0+ means that the 0 came from a term that is positive near the limit point. 0- means that the 0 came from a term that is negative near the limit point. If the sign of the denom near the limit point alternates or is not known. This is shorthand for $lim u/v^2 = \infty $ if $lim u = \infty $ and lim v = 0. This is shorthand for $lim u/v^2^n = \infty $ if $lim u = \infty $ and lim v = 0. This is shorthand for $lim a/u^2 = \infty $ if a>0 and lim u = 0. This is shorthand for $lim a/u^2 = -\infty $ if a<0 and lim u = 0. This is shorthand for $lim a/u^2^n = \infty $ if a>0 and lim u = 0. This is shorthand for $lim a/u^2^n = -\infty $ if a<0 and lim u = 0. This is shorthand for $lim ln u = \infty $ if $lim u = \infty $. This is shorthand for $lim \sqrt u = \infty $ if $lim u = \infty $. This is shorthand for $lim ^n\sqrt u = \infty $ if $lim u = \infty $. The arctan of a large positive (or negative) number is near $\pi /2$ (or $-\pi /2$). The arccot of a large positive number is near 0. The arccot of a large negative number is near $\pi $. The arcsec of a large number is near $\pi /2$. The arccsc of a large number is near 0. None of sin, cos, tan, sec, csc, tan have limits at $\infty $. cosh of a large number x is approximately e^x/2, which is large. sinh of a large number x is approximately e^x/2, which is large. tanh of a large number x is approximately 1, since cosh and sinh are both approximately e^x This is shorthand for $lim ln u = -\infty $ if $lim u = 0$ and $0<u$. The derivative of a constant is zero. The derivative of x with respect to x is 1 The derivative of a sum is the sum of the derivatives. Pull a minus sign out of a derivative. Pull a constant out of a derivative. This is called the power rule. Differentiate a polynomial at once, in one step. Express f'(x) using the d/dx notation for the derivative. This is the definition of the derivative as a limit. Pull a constant out of the denominator. This is called the product rule. Although this is a special case of the quotient rule, memorize it separately. This is called the quotient rule. Use this rule on $\sqrt $, rather than always converting to fractional exponents. Convert roots to fractional exponents in order to differentiate. Use this rule, rather than convert to negative exponents and back again. Use this rule rather than expand |x| by cases. Express f'(x) using the d/dx notation for derivatives. The derivative of sine is cosine. The derivative of cosine is minus the sine The derivative of tangent is secant squared. The derivative of secant is secant tangent. The derivative of cotangent is cosecant squared. The derivative of cosecant is cosecant cotangent. The exponential function is its own derivative. Every exponential function is its own derivative except for a constant ln c. Use this rule to differentiate a power with nonconstant base and exponent. The derivative of ln x is 1/x. ln |x| has the same derivative as ln x but is defined for negative x too. Using this formula is called logarithmic differentiation. Example: d/dx e^(sin x) = e^(sin x) d/dx sin x Example: d/dx 2^(sin x)=(ln 2)2^(sin x) d/dx sin x Example: d/dx ln sin x = (1/sin x)(d/dx sin x) Example: d/dx ln |x^3| = (1/x^3) d/dx x^3 When d/dx ln(cos x) occurs, this rule does it in one step. When d/dx ln(sin x) occurs, this rule does it in one step. If you forget this, differentiate x = tan y and solve for dy/dx. If you forget this, differentiate x = sin y and solve for dy/dx. If you forget this, differentiate x = cos y and solve for dy/dx. If you forget this, differentiate x = cot y and solve for dy/dx. If you forget this, differentiate x = sec y and solve for dy/dx. If you forget this, differentiate x = csc y and solve for dy/dx. Example: d/dx arctan x^2 = d/dx(x^2)/(1+x^4) Example: $d/dx arcsin x^2 = d/dx(x^2)/\sqrt (1-x^4)$ Example: $d/dx arccos x^2 = -d/dx(x^2)/\sqrt (1-x^4)$ Example: $d/dx arccot x^2 = -d/dx(x^2)/(1+x^4)$ Example: $d/dx arcsec x^2 = d/dx(x^2)/(|x^2|\sqrt (x^4-1))$ Example: $d/dx arccsc x^2 = -d/dx(x^2)/(|x^2|\sqrt (x^4-1))$ Example: d/dx (1+x^2)^100 = 100(1+x^2)^99 d/dx x^2 Example: $d/dx \sqrt (1+x^2) = (d/dx x^2)/(2\sqrt (1+x^2))$ Example d/dx sin x^2 = (cos x^2) d/dx x^2 Example: d/dx cos x^2 = -(sin x^2) d/dx x^2 Example: d/dx tan x^2 = (sec^2 x^2) d/dx x^2 Example: d/dx sec x^2 = (sec x^2 tan x^2) d/dx x^2 Example: cot x^2 = -(csc^2 x^2) d/dx x^2 Example: csc x^2 = -(csc x^2 cot x^2) d/dx x^2 Example: d/dx |sin x| = (sin x d/dx sin x)/|sin x| The chain rule applied to any function f, with or without a definition. Introduce a new letter to stand for the selected term. Replace a defined variable by its definition throughout the line. experiment numerically Add the points where $f'(x)=0$ to the list of points considered. Add the endpoints of the interval to the list of points considered. Add the points where $f'(x)$ undefined to the list of points considered. consider limits at open ends reject point outside interval make table of decimal $y$-values for the listed $x$-values. make table of exact $y$-values for the listed $x$-values. choose maximum value(s) from the table. choose minimum value(s) from the table. evaluate derivative in one step solve simple equation evaluate limit in one step eliminate integer parameter For a constant function, the max and min are equal. Evaluate a derivative at once, in one step. Perform algebraic simplification. Solve an equation in one step. Will fail on complicated equations. differentiate both sides of an equation valid for all $t$ in some interval. MathXpert will evaluate the derivative Eliminate a derivative by substituting an expression known to be equal to it. Perform algebraic simplification, collecting, cancelling, ordering, etc. Use various laws to eliminate compound fractions in one step. Put a sum containing fractions over a common denominator and simplify. $ab+ac = a(b+c)$; factors out the greatest explicit common factor Use simple factoring identities to factor as much as possible in one step. Multiply out a product of sums and then collect and/or cancel the terms. Factor out the greatest common divisor of numerator and denominator. Example: write $(x+1)^2 -2x$ as polynomial in x+1, get $(x+1)^2-2(x+1) + 2$. Express in standard polynomial form in the main variable. Example: 3x^2 - 2x + 1 becomes 3(x^2 - 2/3 x + 1/3) Change $x^\onehalf $ to $\sqrt x$ throughout the selected expression. Change fractional exponents to roots throughout the selected expression. Change roots to fractional exponents throughoug the selected expression. Differentiate an identity. The second derivative is the derivative of the derivative. Example: d^3u/dx^3= d/dx d^2u/dx^2 The derivative of the derivative is the second derivative. The derivitive of the n-th derivative is the n+1-st derivative. Calculate a derivative at once, in one step. Compute the value of the current line at a specified point. The integral of 1 with respect to t is just t. The integral of a constant c is ct. Special case of the power rule if we consider t as t to the first power. Pull a constant out of an integral. Pull a minus sign out of an integral. This is called the additivity of the integral. The integral of a difference is the difference of the integrals. This is called the linearity of the integral. This is the power rule for integration. Use this rule instead of always converting to negative exponents. Integrate a polynomial at once, in one step. Don't forget the absolute value; ln |t| is a more natural function than ln t. Multiply out products of sums in the integrand. Example: $\int (t+1)^2 dt = \int t^2+2t+1 dt$ Use this formula rather than expanding |t| by cases. The integral of sine is minus cosine. The integral of cosine is sine. The integral of tangent is -ln cosine, but don't forget the absolute value. The integral of cotangent is ln sine, but don't forget the absolute value. This amazing formula is due to Euler. This formula is almost like the integral of secant, but one sign is different. The derivative of cotangent is minus cosecant squared. If you forget this, remember to write $tan^2$ as $sec^2 - 1$. If you forget this, remember to write $cot^2$ as $csc^2 - 1$. The derivative of cosecant is minus cosecant cotangent. Example: $\int sin 2t dt = -(1/2) cos 2t$ Example: $\int cos 2t dt = (1/2) sin 2t$ Example: $\int tan 2t dt = -(1/2) ln |cos 2t|$ Example: $\int cot 2t dt = (1/2) ln |sin 2t|$ Example: $\int sec 2t dt = (1/2) ln |sec 2t + tan 2t|$ Example: $\int csc 2t dt = (1/2) ln |csc 2t - cot 2t|$ Example: $\int sec^2 2t dt = (1/2) tan 2t$ Example: $\int csc^2 2t dt = -(1/2) cot 2t$ Example: $\int tan^2 2t dt = (1/2) tan 2t - t$ Example: $\int cot^2 2t dt = -(1/2) cot 2t - t$ Example: $\int sec 2t tan 2t dt = (1/2) sec 2t$ Example: $\int csc 2t cot 2t dt = -(1/2) csc 2t$ The exponential function is its own integral, as well as derivative. Example: $\int e^2t dt =(1/2) e^(2t)$ The function e^(-t) is minus its own integral. Example: $\int e^(-2t)dt = -(1/2) e^(-2t)$ Example: $\int e^(t/2)dt = 2 e^(t/2)$ Example: $\int 3^t dt = (1/ln 3) 3^t$8 Example: $\int t^t dt = \int (e^(t ln t) dt$ If you forget this, integrate by parts, taking parts ln t and 1. This is the definition of Erf; the integral has no simpler form. Introduce a new letter for the specified expression. MathXpert will try to find an applicable substitution. Apply this to the equation defining your new variable. Use this when you have calculated du/dx to get the original integral back. Separate out du/dx from the integrand and write the rest as a function of u. This is the substitution rule proper, for which you have been preparing. Replace a defined variable by its definition throughout the current line. Integrate by substitution in one step using the specified expression. Integrate by substitution in one step; let MathXpert choose the substitution. Integrate by parts, using the selected term as the part u to differentiate. Integrate by parts, letting MathXpert choose the parts. This creates an equation which can sometimes be solved for the integral. Transfer the integral to the left side in order to solve for it. Calculate a derivative at once, in one step Integrate by substitution in one step, using the selected term to define u. integrate by substitution in one step, letting MathXpert choose a substitution. Evaluate an integral in one step, if it is not too complicated. This is the derivative form of the Fundamental Theorem of Calculus. This is the integral form of the Fundamental Theorem of Calculus. This is the definition of the symbols on the left side. This is often simpler than ln f(b) - ln f(a) An integral changes sign if its upper and lower limit are exchanged. You will be asked for the point at which to break the integral Example: a definite integral $\int |(t-1)(t+1)| dt$ should be broken at -1 and 1. Specify parameter value, then use approximate numerical integration. Use approximate numerical integration to get a decimal answer. When the upper and lower limits are the same, a definite integral is zero. Converts an improper integral to a limit of proper integrals. If $u$ does not tend to 0 as $t\infty $, then $\int u dt$ from c to $\infty $ diverges. If $u$ does not tend to 0 as $t-\infty $, then $\int u dt$ from $-\infty $ to c diverges. An odd function, integrated over a symmetric interval, yields zero. An even function contributes equally to the integral for plus and minus x. Example: substitute $x = sin \theta $ to integrate $\sqrt (1-x^2)$ Example: substitute $x = tan \theta $ to integrate $\sqrt (1+x^2)$ Example: substitute $x = sec \theta $ to integrate $\sqrt (x^2-1)$ Example: substitute $x = sinh \theta $ to integrate $\sqrt (1+x^2)$ Example: substitute $x = a cosh \theta $ to integrate $\sqrt (x^2-1)$ Example: substitute $x = a tanh \theta $ to integrate $\sqrt (1-x^2)$ You will be asked to enter the definition of x in terms of a new variable Evaluate an integral at once, in one step, if it isn't too complicated. Use this to get rid of $sin^2 t$ in an integral. Use this to get rid of $cos^2 t$ in an integral Use this to integrate an odd power of sin x (also with powers of cos). Use this to integrate an odd power of cos x (also with powers of sin). Use this to integrate an even power of sec x (also with powers of tan). Use this to integrate an even power csc x (also with powers of cot). Use this to integrate an odd power of tan x with power of sec present too. Use this to integrate an odd power of cot x with powers of csc present too. Express $tan^2 x$ in terms of $sec^2 x$ to prepare for u = sec x Express $cot^2 x$ in terms of $csc^2 x$ to prepare for u = csc x $\int sec^n x dx = -1/(n-1) sec^n x tan x + (n-2)/(n-1)\int sec^(n-2) x dx$ $\int csc^n x dx = -1/(n-1) csc^n x cot x + (n-2)/(n-1)\int csc^(n-2) x dx$ This works on any trigonometric integral, but other methods may be simpler. Use this to get rid of 1-cos x in the denominator. Use this to get rid of 1+cos x in the denominator. Use this to get rid of 1-sin x in the denominator. Use this to get rid of 1+sin x in the denominator. Use this to get rid of sin x - cos x in the denominator. Use this to get rid of cos x + sin x in the denominator. Example: (x^2 + 2x + 2)/(x+1) = x + 1 + 1/(x+1) Use all applicable factoring rules to factor the denominator. Factor out the greatest common divisor of numerator and denominator Factor out all repeated factors (greatest common divisor of u and u') Example: x^3-x+1 = (x+1.32472)(x^2 - 1.32472 x + 0.754878) Example: 2x/(x^2-1) = 1/(x-1) + 1/(x+1) Example: x^2 + 4x = (x+2)^2 - 4 Example: $\int 1/(3t-1) dt = (1/3) ln |3t-1|$ Example: $\int 1/(3t+1)^3 dt = -1/6 (3t+1)^2$ Example: $\int 1/(t^2+4)dt=(1/2)arctan(t/2)$ Example: $\int 1/(t^2-4)dt=(1/2)arccoth(t/2)$ Example: $\int 1/(t^2-4)dt=(1/4)ln|(t-2)/(t+2)|$ Example: $\int 1/(4-t^2)dt=(1/2)arctanh(t/2)$ Example: $\int 1/(4-t^2)dt=(1/4)ln|(t+2)/(2-t)|$ Example: $x^2 + 4x = (x+2)^2 - 4$ Example: $\int 1/\sqrt (4-t^2)dt = arcsin(t/2)$ Example: $\int 1/\sqrt (t^2-3)dt)=ln|t+\sqrt (t^2-3)|$ Example: $\int 1/(t\sqrt (t^2-4))dt=(1/2)arccos(t/2)$ That is, integrate by substitution. You specify the substitution. If you forget this, derive it using integration by parts. Use various laws of fractions to eliminate compound fractions in one step. Put sums containing fractions over a common denominator and simplify. ab+ac = a(b+c). Factors out the greatest explicit common factor. Example: x^3 + 2x^2 + x becomes x(x+1)^2 Multiply out products of sums and collect and/or cancel the resulting terms. Solve an equation in one step, if it is not too complicated. Evaluate a limit at once, if MathXpert can do it at all. Integrate by substitution. You will be asked for a substitution. Evaluate an integral in one step, if it isn't too complicated. Example: 3 + c_1 becomes c_2 The integral of sinh is cosh The integral of cosh is sinh The integral of tanh is ln cosh The integral of coth is ln sinh The integral of csch is ln tanh, but it's ln tanh(u/2), not ln tanh u. The integral of sech is arctan of sinh. This converges for |x|<1. Expand $x^k/(1-x)$ in a geometric series. Expand $x^k/(1+x)$ in a geometric series. Formula for the sum of a geometric series starting from an arbitrary term. This converges for all x This is called the binomial series. It converges for |x|<1. Express an infinite series using the first two terms and ... Express an infinite series using the first three terms and ... Example: $1 + x + ... + x^n + ...$ Replace the ... notation with sigma notation One more term of the series will be visible. You will enter how many more terms you want to see. Show the visible part of the series with factorial evaluated. Show the visible part of the series with factorial not evaluated. Show the visible part of the series using decimal coefficients. Do not evaluate the coefficients to decimal form. (a_1-a_0) + (a_2-a_1) + ...= - a_0. The result is a double sum: $(\sum a_n)(\sum b_m) = \sum \sum a_nb_m$ The result is a power series whose coefficients are given by finite sums. The division will be carried out in one step. The result is a double sum: $(\sum a_n)^2 = \sum \sum a_na_m$ The result is a series whose coefficients are defined by a recurrence relation. $\sum u + \sum v = \sum (u + v)$ if the limits of summation are the same. $\sum u - \sum v = \sum (u - v)$ if the limits of summation are the same. The series will be broken into a finite sum plus a new series. Example: change the lower limit from 1 to 0 and subtract the extra term. Example: in a sum involving $x^(n-1)$, add 1 to the index variable. Example: in a sum involving $x^(n+1)$, subtract 1 from the index variable. The index variable can be renamed without changing the value of the series. This law is only valid if the resulting series all converge. Power series and some other series can be differentiated term by term. Power series and some other series can be integrated term by term. Use decimal arithmetic to calculate the sum of a specified number of terms. This is useful if you can expand the derivative in a series. Using a definite integral saves solving for a constant of integration. This is useful if you can expand the integral in a series. Substitute zero (or another value) and solve for the constant. Separate terms with even and odd indices into two different series. Example: $\sum (n-1)/n$ diverges because $lim(n->\infty ,(n-1)/n) = 1$ If $u$ is positive and decreasing, $\sum u$ converges if and only if $\int u dx$ converges. The limit of the ratio of successive terms, if not 1, determines convergence. Limit of the $n$-th root of the $n$-th term, if not 1, determines convergence. Example: $\sum |sin n|/2^n$ converges since $\sum 1/2^n$ converges and $|sin n|< 1$. Example: $\sum ln(n)/n$ diverges since $\sum 1/n$ diverges and $ln(n)/n < 1/n $. If $lim a_n/b_n > 0$ and $a_n>0$ and $b_n>0$ then $\sum a$ converges iff $\sum b$ converges. Replace the $n$-th term of a decreasing series by $2^n$ times the $2^n$-th term. State the result of the test about convergence or divergence. Make the comparison series the current expression so it can be manipulated. State the result of the comparison test as a bound on the original series State the result of the comparison test: the original series is divergent. The harmonic series diverges to infinity. The sum of the reciprocals of the squares is $pi^2/6$. To take the ln of a complex number, first convert to polar form. The ln of a complex number is the ln of the modulus + i times the argument. Since the argument of i (the angle in its polar form) is $\pi /2$ Since the argument of -1 (the angle in its polar form) is $\pi $ Since the argument of a negative number is $\pi $ This famous formula links the trig and complex exponential functions. Halve the argument and take the square root of the modulus. Divide the argument by n and take the n-th root of the modulus. This formula, due to Euler, links several fundamental constants. The complex exponential function is periodic, with period $2\pi i$. To compute a complex power, express it using the exponential function. Express complex sin in terms of sinh Express complex cos in terms of cosh Express complex cosh in terms of cos Express complex sinh in terms of sin Express complex tan in terms of tanh Express complex cot in terms of coth Express complex tanh in terms of tan Express complex coth in terms of cot Fundamental relation between complex exponential and trig functions Definition of complex cos, used in reverse Definition of complex sin, used in reverse This formula defines the hyperbolic cosine function. Definition of cosh, used in reverse. This formula defines the hyperbolic sine function. Definition of sinh, used in reverse. cosh is an even function. sinh is an odd function. The sum of cosh and sinh simplifies to an exponential. The difference of cosh and sinh simplifies to an exponential. This is also the minimum value of cosh. The graph of sinh passes through the origin, since it is an odd function. Express e^x in terms of hyperbolic functions, Express e^(-x) in terms of hyperbolic functions. This identity is analogous to $sin^2 + cos^2 = 1$, but notice the different sign. This identity is analogous to $sin^2 + cos^2 = 1$, but notice the minus sign. This identity is analogous to $cos^2 = 1 - sin^2$, but notice the different sign. This identity is analogous to $sin^2 = 1 - cos^2$, but notice the different sign. This identity is analogous to $1 + tan^2 = sec^2$, but notice the different sign. This identity is analogous to $sec^2 - 1 = tan^2$, but notice the different sign. Definition of the hyperbolic tangent. Definition of tanh in reverse Definition of the hyperbolic cotangent. Definition of coth in reverse Definition of the hyperbolic secant. Definition of sech in reverse. Definition of the hyperbolic cosecant. Definition of csch in reverse. Analogous to $sec^2-tan^2 = 1$, but notice the different sign. Analogous to $tan^2 = sec^2-1$, but notice the different signs. Analogous to $sec^2 = 1 + tan^2$, but notice the different sign. Analogous to the formula for sin(u+v), but the sign is different. Analogous to the formula for cos(u+v), but the sign is different. Analogous to the formula for sin 2u. Analogous to the formula for cos 2u, but the sign is different. Surprise: tanh(ln u) is not as complicated as it looks. The integral of arcsinh is a logarithm of an algebraic function. The integral of arccosh is a logarithm of an algebraic function. The integral of arctanh is a logarithm of a rational function. The derivative of sinh is cosh. The derivative of cosh is sinh. The derivative of tanh is sech^2. The derivative of coth is -csch^2. The derivative of sech u is -sech tanh The derivative of csch is -csch coth The derivative of ln sinh is coth The derivative of ln cosh is tanh Similar to the formula for the derivative of arcsin, but with a sign change. Similar to the formula for the derivative of arccos, but with a sign change. Similar to the formula for the derivative of arctan, but with a sign change. Similar to the formula for the derivative of arccot, but with a sign change. Similar to the formula for the derivative of arcsec, but with a sign change. Similar to the formula for the derivative of arccsc, but with a sign change. sg(x) is the sign of x, 1 if x is positive, -1 if x is negative. sg is an odd function. sg can be expressed in terms of absolute value. Use this inside an integral if the integrand is nonzero. Also works on fractional exponents even/odd. Also works on fractional exponents odd/odd. Use this to get sg in the numerator. sg is not differentiable at zero, but it's constant elsewhere. sg can be integrated directly using this formula. This law is valid only if the integrand is nonzero. If necessary, handle the cases of positive and negative sign separately. Example: sg(3x) = sg(x) Example: sg(ax) = sg(x) if a<0 has been assumed. Example: sg(2x/3) = sg(x) Example: sg(x/a) = sg(x) if a<0 has been assumed. Example: sg(x^3) = sg(x) Example: sg(1/c) = sg(c) Example: sg(3/c) = sg(c) Example: a sg(a) = |a| Example: |a| sg(a) = a The derivative of J_0 is minus J_1. The derivative of J_1 is given in terms of J_0 and J_1. The derivative of J_n is given in terms of J_(n-1) and J_n. The derivative of Y_0 is minus Y_1. The derivative of Y_1 is given in term of Y_0 and Y_1. The derivative of Y_n is given in terms of Y_(n-1) and Y_n. The derivative of I_0 is minus J_1. The derivative of I_1 is given in terms of I_0 and I_1. The derivative of I_n is given in terms of I_(n-1) and I_n. The derivative of K_0 is minus K_1. The derivative of K_1 is given in term of K_0 and K_1. The derivative of K_n is given in terms of K_(n-1) and K_n. Apply user-defined function. ���� | � A�B
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