3. Write down a function f which is analytic in C except for z = 0,1, 2, and which has an essential singularity at z = 0, a removable singularity at z = 1, and a pole of order 5 at z = 2, with residue 14. There are many correct solutions: think about writing f as a sum of three functions, each of which has a single isolated singularity.

Q: Given if (x,y) (0,0) f(x, y) = x4+ y6 %3D if (x, y) = (0,0) where a and b are nonnegative integers.…

A: Given, f(x,y)=xaybx4+y6  if (x,y)≠(0,0)0              if (x,y)=(0,0)       Part A:  a=4 and b=1 then…

Q: deal with Hermite’s equation: y′′−2xy′+2αy =0, −∞< x <∞.  ...........(11.3.13) (e−x2H′…

A:

Q: Q4. Solve the equation z² = 2.

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Q: n 19 The minimum value of function y = x² in the interval [1, 5] is %3D

A: Introduction :

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A: Step 1: To prove ∀k∈N,1k+2k+⋯+nk∈Θ(nk+1), we need to show both 1k+2k+⋯+nk∈O(nk+1) and…

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Q: Write the equation of the degree 3 polynomial P (x), given that a) P(0) = -2, P' (0) = 3, P" (0) =…

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A: The given differential equation is:        That is,     The initial conditions are given by and .

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Q: Draw the hyperbola with the equation: 3x2 − 2y2 + 6x − 8y = 6.

A: Given the equation of the hyperbola : 3x2-2y2+6x-8y=6.

Q: 12: Apply Newton's method with line search for f(x) = x₁-x2 + 2x² + 2x₁x₂ + x2 with a starting point…

A: We have f(X) = x1-x2+2x12+2x1x2+x22

Q: Consider the following lines. r(t) = (3t+ 3, 5t + 1,4t + 4), for - o<t< c∞ R(s) = (s +2, 2s – 1, –…

A: Given that: r(t)=&lt;3t+3, 5t+1, 4t+4&gt;   for -∞&lt;t&lt;∞R(s)=&lt;s+2,2s-1,-3s+7&gt; for…

Q: 4. Verify whether S(x) is a cubic spline x³ x - 1 S(x) = { ²² + ² 17²³ + 2(x − 1)² + 4(x − 1) + 1 −…

A: A function S(x)  is called cubic spline in  if S(x) satisfies the following condition.

Q: 4. Find the residues of each of the following functions at its poles 3. •f(z)= 22 +3 ans : 2²-4 FCZ)…

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Q: Prove L{a coskt}= -at S+a

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Q: Eg. 36 x₁ ~ Belà (M₁, M₂) and X₂ ~ Belä (n₁+1, M₂), the distn of U = √√√××2 If independently,

A: solving only eg-36....   To find the distribution of U, we will use the transformation method. Let Y…

Q: y dy Find using the substitution +5= √√y+2. √x+2 A²/(√y+2-5)(-5√/y+2+y+21) +C 3 ® ²/(√y+2-5)…

A: ∫ydyy+2 Substitute, u+5=y+2 Then , du=12y+2dy ⇒2y+2 du =dy ⇒2u+5 du =dy and y=u+52-2=u2+10u+23

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A: Use the given function.

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Q: Find the absolute Maxima and Minima for f(x) = 2x³ + 21x² + 36x + 10 at [-5, 0].

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Q: The vertex of the quadratic function f(x) = 2(x – 1)2 + 6 is

A: We compare the equation with the vertex equation  y=a(x-h)2+k Comparing we get: h=1 and k=6

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Q: Question 4 a. Let f(x,y) = x² + 2x + 2xy + y² Find f(1, 2) and f(0,0)

A: Note:-Since you asked multiple question, according to our policy we can solve only first question…

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A: The problem concerns interpolating properties of polynomials

Q: 7. Determine an equation of the form y Asin[B(x-C)] +D or y Acos[B(x-C)+D for the following graphs…

A: Here y=AcosBx-C+D⇒A=Amplitude⇒C=Phase shift⇒D=Vertical shift⇒Time period T=2πB

Q: deal with Hermite’s equation: y′′−2xy′+2αy =0, −∞&lt; x &lt;∞.  ...........(11.3.13) 8. Show that if…

A: Given differential equation is: y''-2xy'+2αy=0 Consider: y=∑n=0∞anxn So,…

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.