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What is holomorphic function?
A holomorphic function is a function that is differentiable at every point on its domain. In order for it to be differentiable, it needs to satisfy the Cauchy-Riemann equation properties, such that: f(z) = u(x,y) + iv(x,y) ux = vy vx = -uy If that is so, then f'(z) = ux + ivx Otherwise, if a function doesn't satisfy these conditions, we say that it's not holomorphic. For instance: f(z) = z̅ Test with the following properties: ux = vy vx = -uy z̅ is written as u(x,y) - iv(x,y). Take the partial derivatives of u(x,y) and v(x,y). Then: ux = -vy vx = -(-uy) = uy Since the conditions don't hold, that function is not holomorphic.
A combination of numbers variables and mathematical operations arranged in meaningful order?
variable expression(;
What does join mean in maths?
A join and meet are binary operations on the elements of a POSET, or partially ordered set. A join on a set is defined as the supremum with respect to a partial order on the set, provided it exists. A meet on a set is defined either as the unique infimum with respect to the partial order imposed on the set, if the infimum exists.
How do you multiply fractions with variables?
In order to multiply fractions with variables, factor all numerators and denominators completely. Use the rules for multiplying and dividing fractions, cancel any common factors, and leave your final answer in factored form.
What does it mean to solve an equation?
It means to find what numbers the variables (the letters) must be in order to make the equation a true statement.
What geometric shape starts with the letter j?
The geometric shape that starts with the letter J is a "Jacobian." In mathematics, a Jacobian matrix is a matrix of first-order partial derivatives for a vector-valued function. It is used in multivariable calculus and differential equations to study the relationship between different variables in a system.
What is the formula of bordered Hessian matrix?
The bordered hessian matrix is used for fulfilling the second-order conditions for a maximum/minimum of a function of real variables subject to a constraint. The first row and first column of the bordered hessian correspond to the derivatives of the constraint whereas the other entries correspond to the second and cross partial derivatives of the real-valued function. Other than the bordered entries, the main diagonal of the sub matrix consists of entries for the second partial derivatives. All other entries of the sub matrix off of the main diagonal correspond to all combinations of cross partials. Evaluating the determinant of the bordered hessian will allow one to determine if the function attains its maximum or minimum at the stationary points, which by the way are limited in the fact that they must both satisfy the gradient equations and the constraint.
How can one find higher order derivatives efficiently?
To find higher order derivatives efficiently, one can use the power rule, product rule, quotient rule, and chain rule in calculus. These rules help simplify the process of finding derivatives of functions with multiple variables or functions nested within each other. Additionally, using computer software or calculators can also aid in quickly calculating higher order derivatives.
Why use partial differential equations?
A functional relation can have two or more independent variables. In order to analyse the behaviour of the dependent variable, it is necessary to calculate how the dependent varies according to either (or both) of the two independent variables. This variation is obtained by partial differentiation.
What is a monotonic transformation and how does it impact the relationship between variables in a mathematical function?
A monotonic transformation is a mathematical function that preserves the order of values in a dataset. It does not change the relationship between variables in a mathematical function, but it can change the scale or shape of the function.
What is the relationship between the Green's Theorem Divergence Theorem and Stoke's Theorem?
GREEN'S THEOREM: if m=m(x,y) and n= n(x,y) are the continuous functions and also partial differential in a region 'r' of x,y plane bounded by a simple closed curve c. DIVERGENCE THEOREM: if f is a vector point function having continuous first order partial derivatives in the region v bounded by a closed curve s
Independent vs dependent varibles?
Independent variables are ones that are not impacted by outside forces or other factors. Dependent variables are the ones that are impacted by other elements and need other elements in order to function.
What are register variables What are the advantage of using register variables?
Asks the compiler to devote a processor register to this variable in order to speed the program's execution. The compiler may not comply and the variable looses it contents and identity when the function it which it is defined terminates.
What if the fourth derivative of a polynomial is zero?
There are many things that can be said about a polynomial function if its fourth derivative is zero, but the main thing you can know about this function from this information is that its order is 3 or less. Consider an nth order polynomial with only positive exponents: axn + bxn-1 + ... + cx2 + dx + e As you derive this function, its derivatives will eventually be equal to zero. The number of derivatives that are nonzero before they all become zero can tell you what order the polynomial function was. Consider an example, y = x4. y = x4 y' = 4x3 y'' = 12x2 y''' = 24x y(4) = 24 y(5) = 0 The original polynomial was of order 4, and its derivatives were nonzero up until its fifth derivative. From this, you can generalize to say that any function whose fifth derivative is equal to zero is of order 4 or less. If the function was of higher order than 4, its derivatives would not become zero until later. If the function was of lower order than 4, its fifth derivative would still be zero, but it would not be the first zero-valued derivative. So this experimentation yielded a rule that the first zero-valued derivative is one greater than the order of the polynomial. Your problem states that some polynomial has a fourth derivative that is zero. Our working rule states that this polynomial can be of highest order 3. So, your polynomial can be, at most, of the form: y = ax3 + bx2 + cx + d Letting the constants a through d be any real number (including zero), this general form expresses any polynomial that will satisfy your condition.
What are the three internal variables that are concerned with homeostasis?
The three internal variables that are concerned with homeostasis are body temperature, blood glucose level, and blood pH. These variables must be regulated within a narrow range in order for the body to function properly and maintain equilibrium.
What is the basic meaning of the word partial?
The meaning of the word partial is not all there. For instance if you order an outfit online and you first get the shirt it would be a partial shipment of your order.
What is the resulting ordered pair if the value of the independent variable is 2?
In order to answer that, I would have to know how the dependant and independant variables are related ... how one depends on the other one. That's called the 'function', or the 'equation in two variables'. It's probably right there, near where you copied the question from.
