According to complex analysis assignment help experts, Complex Analysis developed as a separate field of study under mathematics in the late 19th century. Traditionally, it deals with the functions of the variables and complex numbers. It also involves the study of their derivatives, manipulation and other physical properties through difficult complex analysis assignments. Nowadays many students of mathematics pursue their specialization on the complex analysis which promises the students a promising future. From algebraic geometry to nuclear technology, it has profound usefulness in every field. When complex analysis is evolving as one of the fastest growing fields, MyAssignmenthelp.com has brought the complex analysis assignment help and complex analysis homework help services for the benefit of the students.
Understanding complex analysis
Though it developed as a distinct branch in the 19th and 20th century, Complex analysis is referred to as a classical branch of Mathematics. With the worthy contributions from mathematicians like Euler, Gauss, Cauchy and Riemann, it became an important part of the discipline. Today, it is a popular and powerful tool which provides practical solutions to the scholars, say the complex analysis assignment help experts.
The field mostly deals with the complex functions on which the students receive complex analysis assignments. These forms of functions have an independent variable and a dependent variable which can be separated into real and imaginary parts. In a complex function, the domain and the range all are the parts subsets of the given complex plane known as Argand Plane (R2), say the complex analysis assignment experts.
Here is an example from a sample complex analysis assignment. If a given complex number z equals to x + iy, then its real and imaginative parts define an element of R2 (x, y).
If w= ƒ (z)
Then, u (x, y) + i v (x, y)
As x, y ∈ R and u (x, y) & v (x, y) are real valued functions which are the components of ƒ (z).
This is the primary concept on which all complex analysis assignments are based. It is gradually introduced into the various segments of real functions into the domain. It may range from exponential functions to trigonometric functions on which the students need to prepare their complex analysis assignments.
Theorems of complex analysis
The scope of the complex analysis is vast and elaborate. The main segments of complex analysis as told by complex analysis assignment help experts are:
- Holomorphic Functions
- Complex Functions
- Cauchy’s Theorem
- Power Series
- Taylor and Laurent series
- Isolated Singularities and Residue Theorem
- Discrete Applications
- Residue Theorem
After consulting the eminent complex analysis assignment help experts, here are the most prominent principles, theorems and functions of complex analysis.
Cauchy theorem
According to the complex analysis assignment help experts, it is the central theorem of the complex analysis. It states that if γ0 and γ1 are the closed paths in the region of G ⊆ C where γ0(t) and γ1(t) is 0 ≤ t ≤ 1 then the γ0 is G- homotopic to γ1 and there exists a continuous function h: [0, 1]2 -->G
Such that
h (t, 0) = γ0(t)
h (t, 1) = γ1(t)
and
h (0, s) = h (1, s)
This function h (t, s) is known as homotopy. For each fixed s, a homotopy h (t, s) is a path described by t, and as s goes from 0 to 1, these closed paths continuously transform from γ0 and γ1. To know more, call the complex analysis assignment help experts.
Differentiability and Holomorphicity
Some complex functions exhibit peculiar behaviors within its range of limits, say the complex analysis assignment help experts. These functions have an additional dimension which makes these differences become apparent. An example from sample complex analysis assignment:
If ƒ : G-->C is a complex function and z’ is any interior point of G, then the derivative of ƒ at the point z’ is defined as
assuming that the limit exists.In the above function, ƒ is differential at the point z’. If it is differential at all the points in the given plane centered at z’ then the ƒ function will be called holomorphic at z’. If the function is differentiable in the whole complex plane, then it will be referred as entire. Receive complex analysis assignment help to know more.
Isolated singularity
According to complex analysis assignment experts, if ƒ is holomorphic in the D [z0, R] and R > 0 but not at z = z0, in that case z0 will be called isolated singularity of ƒ. The singularity z0 can be of three types.
- It will be called removable, if another function g is holomorphic in D [z0, R] such that ƒ = g in the same D [z0, R].
- It will be called a pole if
- It will be called essential if z0 is neither a pole nor removable
Take complex analysis assignment help and complex analysis homework help to know more.
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Complex analysis problems and solutions
Complex analysis assignment sample 1
Assume U be the unit disk in the complex plane C, U = {z ∈ C : |z| < 1}. Similarly, let ƒ be an analytic function in the neighborhood of the closure of U. If ƒ is real on all the boundary of U, show that ƒ must be constant.
Solution
According to the hypothesis, it implies that Im ƒ (eiÆŸ) = 0 for all ÆŸ ∈ R. Since ƒ is holomorphic in a neighborhood Ω of U, the series
and it converges uniformly on any compact subset of Ω.
The unit circle T = {z : |z| = 1} = { eiÆŸ : θ ∈ R } is the compact subset which has the series
So, using the co-efficient as an = cn + ibn, where cn and bn ∈ R then we get
an einÆŸ - ( cn + ibn ) [cos (nÆŸ) + i sin (nÆŸ)] – [cn cos (nÆŸ) – b sin (nÆŸ)] + i [cn sin (nÆŸ) + b cos (nÆŸ)]
Thus, with the hypothesis, the series is
It converges uniformly to zero for all where ∈ [0, 2π]
Therefore, with the exception of c0, we get cn - bn - 0 for all n
Hence, the result ƒ= c0.
Complex analysis assignment sample 2
Using the contour integration and the residue method to evaluate the integral
Solution
Denoting the integral by I which is even,
In the closed contour, |R – R U [-R, R], and the trace of R is the set
{R eiÆŸ : 1< ÆŸ < π }. When R>1 then I inside the region bounded by R.
it is holomorphic inside and on R, except for the double pole at z=i
The residue of it is
Hence,
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