DirichletEta

✖
`DirichletEta`

✖

DirichletEta[s]

gives the Dirichlet eta function .

Details

The Dirichlet eta function is also known as the alternating zeta function.
DirichletEta is a mathematical function, suitable for both symbolic and numeric manipulation.
For , the Dirichlet eta function is defined as $TemplateBox[{s}, DirichletEta]=sum_(n=0)^infty((-1)^n)/((n+1)^s)$ .
For certain special arguments, DirichletEta automatically evaluates to exact values.
DirichletEta is an entire function with branch cut discontinuities.
DirichletEta can be evaluated to arbitrary numerical precision.
DirichletEta automatically threads over lists.
DirichletEta can be used with Interval and CenteredInterval objects. »

Examples

open allclose all

Basic Examples (3)Summary of the most common use cases

Plot over the reals:

In[1]:=1

✖

https://wolfram.com/xid/0hyvexhtj8dm5nbv-emeds0

Out[1]=1

Visualize in the complex plane:

In[1]:=1

✖

https://wolfram.com/xid/0hyvexhtj8dm5nbv-dbngg6

Out[1]=1

Compute a special value:

In[1]:=1

✖

https://wolfram.com/xid/0hyvexhtj8dm5nbv-cyxafg

Out[1]=1

Scope (6)Survey of the scope of standard use cases

DirichletEta is neither non-decreasing nor non-increasing:

In[1]:=1

✖

https://wolfram.com/xid/0hyvexhtj8dm5nbv-2ra8g

Out[1]=1

DirichletEta is not injective:

In[1]:=1

✖

https://wolfram.com/xid/0hyvexhtj8dm5nbv-c9npzh

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0hyvexhtj8dm5nbv-b5buvp

Out[2]=2

DirichletEta is neither non-negative nor non-positive:

In[1]:=1

✖

https://wolfram.com/xid/0hyvexhtj8dm5nbv-dvzykj

Out[1]=1

DirichletEta is neither convex nor concave:

In[1]:=1

✖

https://wolfram.com/xid/0hyvexhtj8dm5nbv-l0srvu

Out[1]=1

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

In[1]:=1

✖

https://wolfram.com/xid/0hyvexhtj8dm5nbv-bq4opg

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0hyvexhtj8dm5nbv-cmdnbi

Out[2]=2

Or compute average-case statistical intervals using Around:

In[3]:=3

✖

https://wolfram.com/xid/0hyvexhtj8dm5nbv-cw18bq

Out[3]=3

Compute the elementwise values of an array:

In[1]:=1

✖

https://wolfram.com/xid/0hyvexhtj8dm5nbv-thgd2

Out[1]=1

Or compute the matrix DirichletEta function using MatrixFunction:

In[2]:=2

✖

https://wolfram.com/xid/0hyvexhtj8dm5nbv-o5jpo

Out[2]=2

Properties & Relations (1)Properties of the function, and connections to other functions

Verify the interrelationship among the DirichletEta, DirichletLambda and Zeta functions:

In[1]:=1

✖

https://wolfram.com/xid/0hyvexhtj8dm5nbv-cr76jt

Out[1]=1

Top