Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.

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2
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1answer
42 views

Limit problem calculating directional derivative

Given $f(x,y)=1-x^2-y^2$, find the directional derivative at the point $(x_0,y_0)$ in the direction of the unit vector $\vec u$. I am experiencing some strange behavior with the Limit function. I ...
2
votes
2answers
158 views

Two ways to speed up slow Integrate[]s? (Can't do either)

I run into problems like this all the time on the complicated integrands I pass to Integrate. The following code works but takes a veeery looong time: ...
2
votes
3answers
152 views

How can I get the exact value minimum of this function?

I want to find min of the function $$\frac{1}{\sqrt{2 x^2+\left(3+\sqrt{3}\right) x+3}}+\frac{1}{\sqrt{2 x^2+\left(3-\sqrt{3}\right) x+3}}+\sqrt{\frac{1}{3} \left(2 x^2+2 ...
2
votes
1answer
93 views

Limits and transformation rules

I can calculate this limit: Limit[(c^(1 - g) - 1)/(1 - g), g -> 1] and obtain the correct value: Log[c] but if I try ...
2
votes
1answer
2k views

Laurent series expansion

Can someone share how to find a Laurent series expansion of $$f(z)=\frac{1}{(z^2-1)(z^2-4)}$$ centered at zero on the annular disk $1<|z|<2$?
2
votes
1answer
871 views

How to find all roots and maxima and minima and graph them

I need to find all the roots of this function. I know it is simple to find by hand, however, I wish to learn how to do it so I can apply it later. The problem is that it only gives me one root. ...
2
votes
1answer
858 views

Finding residues of multi-dimensional complex functions

Say I have a function $f$ of $n$ complex variables, $\{ z_i \}_{i=1}^{i=Nc}$. And then I want to contour integrate the expression such that for each $z_i$ its an integration on an unit circle about ...
2
votes
1answer
332 views

Assumptions with D

I want to give assumptions for the D function. Say I want to calculate $\frac{d|x|}{dx}$ for $x>0$, which is 1. I write ...
2
votes
1answer
237 views

Checking if first derivative is zero at a given point

I have a function V[a_, h_, tau_] := (Sqrt[3]/2) h (Sqrt[3] a + h/3 tau)^2 now I want to check, whether its first derivative is zero in a given point. Say for ...
2
votes
1answer
97 views

Numerically integrate and plot $f(x;p)$ for a range of the parameter $p$

Suppose we have an integral of the form $$I = \int_a^b dx \, \sin(px), \tag{1}$$ where I've take $f(x;p) = \sin (px)$ to be specific. I want to plot this integral for a range of values of the ...
2
votes
1answer
93 views

How to avoid complex terms in the results given by Integrate?

As shown by In1, integrating Sqrt[x^2 - 1]/(a + x^2) (1/x) returns a result with a complex terms; however, after I make a ...
2
votes
2answers
375 views

Integration of piecewise function

I'm trying to understand why Mathematica is not evaluating a piecewise function, while it's able to evaluate each of the regions separately. This fails: ...
2
votes
1answer
162 views

Check for holomorphy of a function

Given a (rather complicated) function H(z), what is the best approach to check symbolically whether it is holomorphic? What I tried is checking explicitly the ...
2
votes
3answers
143 views

Converting HeavisideTheta[]s and Sign[]s functions to a single Piecewise[]

I have the product of some Heaviside functions or sign functions, a simple example could be: HeavisideTheta[q^2 - 4]*Sign[q - 3] Another slightly more ...
2
votes
1answer
306 views

Prove (or check) an expression is positive given constraints on variables?

I'm trying to figure out whether an expression is always positive given positive parameters. When the expression is complicated, I can't do this by eye. Is there any way to make Mathematica prove ...
2
votes
2answers
216 views

Another fractional part related question

Having known the fact that the evaluation of the integrals containing fractional parts often contains errors, I wonder if the integral below possibly evaluates to $0.7$ or this is just a coincidence ...
2
votes
2answers
285 views

integrating slightly complicated expression which includes Abs

I have been waiting for Mathematica to give me something for the following integral, errors welcome, but it has been "running" for almost 30 minutes now. ...
2
votes
1answer
340 views

Calculus 1 Related Rates -Total Derivative issues

I'm trying to do part a) of this calculus 1 problem in Mathematica V9: http://tutorial.math.lamar.edu/Classes/CalcI/RelatedRates.aspx#Deriv_Rates_Ex4 Warning noob code below! ...
2
votes
1answer
212 views

How can I evaluate surface integral in Mathematica?

$$\iint_{S^+}x^3dydz $$where $S$ bottom part of $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ $S^+$ - outer side of $S$ $$\iint_{S^+}P(x,y,z)dydz = \iint_{S}P\cos\alpha dS,\ \; normal : ...
2
votes
2answers
254 views

Using RegionPlot3D to make a cone under restrictions

I have to plot a cone that satisfies this statement: Use the RegionPlot3D command to draw a picture of a cone with its point at the origin. The other end of the cone is a circle with radius 5 ...
2
votes
1answer
156 views

Definite Integral over Bessel Function

Hello I am interested in evaluating the following integral. ...
2
votes
1answer
165 views

Double integral changing the limits

This integral does not evaluate: Integrate[ Integrate[1/(E^((a - c)^2 + (b - c)^2) ), {c, 0, 1}], {b, -Infinity, Infinity}] However, reversing the integrals ...
2
votes
2answers
167 views

How to calculate the analytical result of this singular integral? [duplicate]

I'm trying to solve this integral: $I(a,b)=\displaystyle{\int_1^{\infty }\dfrac{e^{-ax}}{\sqrt{x^2-1} (1+x\sqrt{1-b^2}) (x-1/\sqrt{1-b^2})} \, dx} \,\, \,\,(a \;\textrm{real>0} \, , 0<b<1)$ ...
2
votes
2answers
240 views

Calculate information entropy integral in infinite square well problem

In the context of information theory, entropy is a measure of uncertainty of a random variable. In quantum mechanics, the uncertainty principle states that $\Delta x\Delta k \ge 1/2$. The same can be ...
2
votes
2answers
163 views

Limit of integral gives incorrect output

I'm trying to evaluate $\lim_{e\to 0} \, \frac{i}{e}\int_{\pi }^0 \frac{1-\exp (i e \exp (i \theta ))}{\exp (i \theta )} \, d\theta$ with Mathematica 9.0.1.0 on OS X. However, I get "Undefined" for ...
2
votes
1answer
99 views

Use Mathematica to determine the falling law

We have a one-variable equation $\rho(R)$ where ρ = (14656.4+277.526*R^2)/(45.9225+R^2)^{5/2} + 0.370036/(R*(0.25+R)^3) This equations describes the evolution of ...
2
votes
1answer
171 views

A question about derivatives (cubics)

Given an equation: f(z) = a (x + I y)^3 + b (x + I y)^2 + c (x + i y) + d Which can be re-written as: ...
2
votes
1answer
249 views

Integrate[(1 + x/n)^n*Exp[-x], {x, 0, Infinity}]

I evaluated Integrate[(1 + x/n)^n*Exp[-x], {x, 0, Infinity}] . thinking the answer should be approximately $\sqrt{\pi n/2}$. Mathematica gave me ...
2
votes
1answer
173 views

How can I calculate a limit with a free variable?

For example, when I evaluate Limit[(1 + x^n + (x^2/2)^n)^(1/n), n -> Infinity] Mathematica does not output any result. When I evaluate ...
2
votes
1answer
144 views

Evaluating a surface integral

I'm trying to compute the integral $$\int_S x^2+z^2\,{\rm d}S,$$where $S$ is the surface $$S\colon~ \frac{x^2}{2} + \frac{y^2}{3} + \frac{z^2}{2} = 1, \quad y \geq 0.$$ One possible parametrization ...
2
votes
1answer
52 views

Meaning of Pattern at derivatives of arbitrary functions

I want to derivate some general functions like D[F[x[r_, p_], y[r_, p_]], r] or this simpler one D[F[G[x_]], x] . When I ...
2
votes
1answer
99 views

Missing zero in the CoefficientList

The following list has a length only 10 (not 11) CoefficientList[Series[ArcTan[x]/(1-x^2),{x,0,10}],x] {0, 1, 0, 2/3, 0, 13/15, 0, 76/105, 0, 263/315} It should ...
2
votes
2answers
155 views

Calculating a Volume Integral limited by two z planes and a given conical surface [closed]

I have been given this question to integrate a function over a region, and I have been at it for 6 days now and I can not for the life of me come up with a solution because the integral is way too ...
2
votes
1answer
114 views

How to get a more compact form of this probability calculation?

Inspired by the probability calculation here, I am trying to solve a little general one: $$\mathbb{P} (\sum_{i=1}^{m-1} A_i + \sum_{i=1}^{m} S_i < L < \sum_{i=1}^{m} A_i + \sum_{i=1}^{m+1} ...
2
votes
1answer
103 views

Hilbert transform of density of states [closed]

I am trying to evaluate analytical integration over density of states, for example G(Z)=int[f(x)/z-x,{x,x1,x2}] , where ...
2
votes
2answers
139 views

Graphical solution of a a two-variable system containing an equation and an inequality

I'm currently studying atmospheric-gravity waves and their dispersion relations. I want to have look at values of $(x,y)$ that solve the following system : \begin{equation}\label{}(1) \left\{ ...
2
votes
1answer
703 views

Time derivative of unit vector in spherical coordinates

Is it possible to take a time derivative of a vector given in some curvelinear coordinate system (i.e. spherical)? Mathematica would need to take into account the time dependence of the basis vectors. ...
2
votes
2answers
95 views

ConditionalExpression Limiting Domain

I came across a problem when evaluating the following integral: $\int_0^t \sqrt{9 x^4+1} \, dx$ Now, when I evaluate that, I get ...
2
votes
1answer
896 views

How to take the derivative w.r.t. an arbitrary function?

EDIT: The answer posted by @Jens works for built-in functions only. However I'm starting to wonder if this is related to my Mathematica install somehow. I have version 7. Here is an example in the ...
2
votes
1answer
414 views

How can I handle curve singularity in this NIntegrate integration?

Yesterday I asked a question about the non converging integral. Woods told me that it is due to the function which has a singularity along a line which passes through the integration region. (Why ...
2
votes
1answer
229 views

Computing derivatives of a moment generating function

Dear Mathematica users, I'm trying to compute higher order derivatives of a moment generating function and then evalutate them in 0 (in order to get some moment conditions for a GMM estimation). ...
2
votes
1answer
304 views

Asymptotic Rate of Growth

How can I calculate the Asymptotic Rate of growth of a function, for instance like: $X^3 - X^2 - X -1$ EDIT: For instance, as you can see in this graph, after the 1200 the function approximates to ...
2
votes
3answers
285 views

Need help evaluating definite integral to a function of Y

Suppose $Y = \sqrt{2T}\cos(U)$, $ 0 \le u \le \pi $, and $ 0 \le \cos^{-1}(\frac{y}{\sqrt {2t}}) \le \pi ) $, so $ -1 \le \frac{y}{\sqrt{2t}} \le 1 $, with all $ \mathbb{R}$. The iterated integral ...
2
votes
2answers
61 views

Extrema of a function of three variables

To identify and classify the critical points of the function $f(x,y,z)=x^3+xz^2-3x^2+y^2+2z^2$, I used the Hessian matrix method. ...
2
votes
1answer
83 views

Implicit derivatives in one line. How do I do it?

I have this little script to derive implicitly $PrePrint = # /. {D[y_, x_, NonConstants -> {y_}] :> y'[x]} &; Example: ...
2
votes
2answers
117 views

Dealing with Norm, Complex situation not desired

I am working on a notebook for my calculus students and am dealing frequently with the norm of vector valued functions. I always seem to run into this kind of situation. ...
2
votes
1answer
76 views

Forcing Exact Outputs for Integrate [closed]

I'm trying to compute the following triple integral: ...
2
votes
1answer
481 views

Inverse gradient operator

I found a nice paper about inverse vector operators here. I have successfully implemented a Mathematica function for most of them, however I can't figure out how to do inverse gradient (page 7 in the ...
2
votes
1answer
384 views

Calculating the area under a curve

How can I calculate the area under a curve after plotting discrete data as per below? ...